795 lines
22 KiB
Markdown
795 lines
22 KiB
Markdown
## 3 номер - Д 847
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#семестр_1 #высшая_математика
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### Пример:
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$y= \dfrac{x}{(1-x)^{2}(1+x)^{3}}$
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$y'=?$
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### Решение:
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$\dfrac{x}{(1-x)^{2}(1+x)^{3}}$
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$y= \dfrac{u}{v}; y'= \dfrac{u'v-uv'}{v^{2}};$
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$u=x; v=(1-x)^{2}(1+x)^{3}$
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$u'=1;$
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$v=ab; v'= a'b+b'a; a=(1-x)^{2}; b=(1+x)^{3};$
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$a'=-2(1-x);$
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$b'=3(1+x)^{2}$
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$v'=(−2(1−x))(1+x)^{3}+(1−x)^{2}3(1+x)^{2}$
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$y'=\dfrac{(1-x)^{2}(1+x)^{3}-x[(-2(1-x))(1+x)^{3}+(1-x)^{2}3(1+x)^{2}]}{(1-x)^{4}(1+x)^{6}}=\dfrac{4x^{2}-x+1}{(1-x)^{3}(1+x)^{4}}$
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## 7 номер - Д 851
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### Пример:
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$y=x+\sqrt{ x }+\sqrt[ 3 ]{ x }$
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$y'=?$
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### Решение:
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$y=x+x^{\frac{1}{2}}+x^{\frac{1}{3}}$
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$(x)'=1;$
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$(x^{\frac{1}{2}})'=\dfrac{1}{2}x^{-\frac{1}{2}};$
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$(x^{\frac{1}{3}})'=\dfrac{1}{3}x^{-\frac{2}{3}};$
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$y'=1+\dfrac{1}{2}x^{-\frac{1}{2}}+\dfrac{1}{3}x^{-\frac{2}{3}}=1+\dfrac{1}{2\sqrt{x}}+\dfrac{1}{3\sqrt[3]{x^{2}}}$
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## 11 номер - Д 855
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### Пример:
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$y=(1+x)\sqrt{ 2+x^{3} }\sqrt[ 3 ]{ 3+x^{3} }$
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$y'=?$
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### Решение:
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$(1+x)\sqrt{2+x^3}\sqrt[3]{3+x^3}$
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$y=abc; y'=a'bc+ab'c+abc';$
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$a=1+x;\ b=\sqrt{2+x^3};\ c=\sqrt[3]{3+x^3}$
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$a'=1;$
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$b=(2+x^3)^{\frac{1}{2}};\ b'=\dfrac{1}{2}(2+x^3)^{-\frac{1}{2}}(3x^2)=\dfrac{3x^2}{2\sqrt{2+x^3}};$
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$c=(3+x^3)^{\frac{1}{3}};\ c'=\dfrac{1}{3}(3+x^3)^{-\frac{2}{3}}(3x^2)=\dfrac{x^2}{\sqrt[3]{(3+x^3)^2}};$
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$y'=\sqrt{2+x^3}\sqrt[3]{3+x^3}+(1+x)\dfrac{3x^2}{2\sqrt{2+x^3}}\sqrt[3]{3+x^3}+(1+x)\sqrt{2+x^3}\cdot\dfrac{x^2}{\sqrt[3]{(3+x^3)^2}}$
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## 15 номер - Д 859
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### Пример:
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$y=\dfrac{1}{\sqrt{ 1+x^{2} }(x+\sqrt{ 1+x^{2} })}$
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$y'=?$
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### Решение:
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$\dfrac{1}{\sqrt{1+x^2}(x+\sqrt{1+x^2})}$
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$y=\dfrac{u}{v};\ y'=\dfrac{u'v-uv'}{v^2};$
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$u=1;\ v=\sqrt{1+x^2}(x+\sqrt{1+x^2})$
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$u'=0;$
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$v=ab;\ v'=a'b+b'a;$
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$a=\sqrt{1+x^2};\ b=x+\sqrt{1+x^2}$
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$a=(1+x^2)^{\frac{1}{2}};\ a'=\dfrac{1}{2}(1+x^2)^{-\frac{1}{2}}2x=\dfrac{x}{\sqrt{1+x^2}};$
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$b'=1+a'=1+\dfrac{x}{\sqrt{1+x^2}};$
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$v'=\dfrac{x}{\sqrt{1+x^2}}(x+\sqrt{1+x^2})+\sqrt{1+x^2}(1+\dfrac{x}{\sqrt{1+x^2}})=\dfrac{(x+\sqrt{1+x^2})^2}{\sqrt{1+x^2}}$
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$y'=-\dfrac{\dfrac{(x+\sqrt{1+x^2})^2}{\sqrt{1+x^2}}}{(\sqrt{1+x^2}(x+\sqrt{1+x^2}))^2}=-\dfrac{1}{(1+x^2)^{\frac{3}{2}}}$
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## 19 номер - Д 863
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### Пример:
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$y=(2-x^{2})\cos x + 2x \sin x$
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$y'=?$
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### Решение:
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$(2-x^2)\cos x+2x\sin x$
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$y=u+v;\ y'=u'+v';$
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$u=(2-x^2)\cos x;\ v=2x\sin x$
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$u=ab;\ u'=a'b+b'a;$
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$a=2-x^2;\ b=\cos x;$
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$a'=-2x;$
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$b'=-\sin x;$
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$u'=(-2x)\cos x+(2-x^2)(-\sin x)=-2x\cos x-(2-x^2)\sin x;$
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$v=ab;\ v'=a'b+b'a;$
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$a=2x;\ b=\sin x;$
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$a'=2;$
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$b'=\cos x;$
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$v'=2\sin x+2x\cos x;$
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$y'=(-2x\cos x-(2-x^2)\sin x)+(2\sin x+2x\cos x)=x^2\sin x$
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## 22 номер - Д 866
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### Пример:
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$y=\sin[\sin(\sin x)]$
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$y'=?$
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### Решение:
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$\sin[\sin(\sin x)]$
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$y=\sin u;\ y'=\cos u\cdot u';$
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$u=\sin(\sin x)$
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$u=\sin v;\ u'=\cos v\cdot v';$
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$v=\sin x$
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$v'=\cos x;$
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$u'=\cos(\sin x)\cos x;$
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$y'=\cos(\sin(\sin x))\cos(\sin x)\cos x$
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## 23 номер - Д 867
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### Пример:
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$y=\dfrac{\sin ^{2}x}{\sin x^{2}}$
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$y'=?$
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### Решение:
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$\dfrac{\sin^2x}{\sin x^2}$
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$y=\dfrac{u}{v};\ y'=\dfrac{u'v-uv'}{v^{2}};$
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$u=\sin^2x;\ v=\sin x^2$
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$u=(\sin x)^2;\ u'=2\sin x\cos x;$
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$v=\sin(x^2);\ v'=\cos(x^2)\cdot2x=2x\cos(x^2);$
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$y'=\dfrac{(2\sin x\cos x)\sin(x^2)-\sin^2x\cdot(2x\cos(x^2))}{\sin^2(x^2)}$
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## 26 номер - Д 880
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### Пример:
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$y=e^{x}(1+\cot \dfrac{x}{2})$
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$y'=?$
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### Решение:
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$e^x(1+\cot\frac{x}{2})$
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$y=ab;\ y'=a'b+b'a;$
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$a=e^x;\ b=1+\cot\frac{x}{2}$
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$a'=e^x;$
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$b'=0+(\cot\frac{x}{2})';$
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$(\cot u)'=-\dfrac{1}{\sin^2u}\cdot u';$
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$u=\dfrac{x}{2};\ u'=\dfrac{1}{2};$
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$(\cot\frac{x}{2})'=-\dfrac{1}{\sin^2(\frac{x}{2})}\cdot\dfrac{1}{2}=-\dfrac{1}{2\sin^2(\frac{x}{2})};$
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$y'=e^x(1+\cot\frac{x}{2})+e^x(-\dfrac{1}{2\sin^2(\frac{x}{2})})=e^x(1+\cot\frac{x}{2}-\dfrac{1}{2\sin^2(\frac{x}{2})})$
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## 27 номер - Д 881
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### Пример:
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$y=\dfrac{\ln3\cdot \sin x + \cos x}{3^{x}}$
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$y'=?$
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### Решение:
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$\dfrac{\ln3\sin x+\cos x}{3^x}$
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$y=\dfrac{u}{v};\ y'=\dfrac{u'v-uv'}{v^2};$
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$u=\ln3\sin x+\cos x;\ v=3^x$
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$u'=\ln3\cos x-\sin x;$
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$v'=3^x\ln3;$
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$y'=\dfrac{(\ln3\cos x-\sin x)3^x-(\ln3\sin x+\cos x)3^x\ln3}{(3^x)^2}$
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$y'=\dfrac{\ln3\cos x-\sin x-(\ln3\sin x+\cos x)\ln3}{3^x}$
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$y'=\dfrac{-\sin x-(\ln3)^2\sin x}{3^x}=-\dfrac{(1+(\ln3)^2)\sin x}{3^x}$
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## 30 номер - Д 884
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### Пример:
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$y=(\dfrac{a}{b})^{x}(\dfrac{b}{x})^{a}(\dfrac{x}{a})^{b}$
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$y'=?$
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### Решение:
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$(\dfrac{a}{b})^x(\dfrac{b}{x})^a(\dfrac{x}{a})^b$
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$y=uvw;\ y'=u'vw+uv'w+uvw';$
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$u=(\dfrac{a}{b})^x;\ v=(\dfrac{b}{x})^a;\ w=(\dfrac{x}{a})^b$
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$u=(\dfrac{a}{b})^x=e^{x\ln(\frac{a}{b})};\ u'=(\dfrac{a}{b})^x\ln(\dfrac{a}{b});$
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$v=(\dfrac{b}{x})^a=b^a x^{-a};\ v'=-ab^a x^{-a-1}=-\dfrac{a}{x}(\dfrac{b}{x})^a;$
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$w=(\dfrac{x}{a})^b=x^b a^{-b};\ w'=bx^{b-1}a^{-b}=\dfrac{b}{x}(\dfrac{x}{a})^b;$
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$y'=(\dfrac{a}{b})^x(\dfrac{b}{x})^a(\dfrac{x}{a})^b(\ln(\dfrac{a}{b})-\dfrac{a}{x}+\dfrac{b}{x})$
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## 34 номер - Д 888
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### Пример:
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$y=\ln(\ln ^{2}(\ln ^{3}x))$
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$y'=?$
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### Решение:
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$\ln(\ln^2(\ln^3x))$
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$y=\ln u;\ y'=\dfrac{u'}{u};$
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$u=\ln^2(\ln^3x)=(\ln(\ln^3x))^2$
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$u=v^2;\ u'=2vv';$
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$v=\ln(\ln^3x)$
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$v=\ln t;\ v'=\dfrac{t'}{t};$
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$t=\ln^3x=(\ln x)^3$
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$t=w^3;\ t'=3w^2w';$
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$w=\ln x;\ w'=\dfrac{1}{x};$
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$t'=3(\ln x)^2\cdot\dfrac{1}{x};$
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$v'=\dfrac{3(\ln x)^2\cdot\frac{1}{x}}{(\ln x)^3}=\dfrac{3}{x\ln x};$
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$u'=2\ln(\ln^3x)\cdot\dfrac{3}{x\ln x}=\dfrac{6\ln(\ln^3x)}{x\ln x};$
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$y'=\dfrac{\frac{6\ln(\ln^3x)}{x\ln x}}{(\ln(\ln^3x))^2}=\dfrac{6}{x\ln x\cdot\ln(\ln^3x)}$
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## 38 номер - Д 896
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### Пример:
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$y=x\ln(x+\sqrt{ 1+x^{2} })-\sqrt{ 1+x^{2} }$
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$y'=?$
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### Решение:
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$x\ln(x+\sqrt{1+x^2})-\sqrt{1+x^2}$
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$y=u-v;\ y'=u'-v';$
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$u=x\ln(x+\sqrt{1+x^2});\ v=\sqrt{1+x^2}$
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$u=ab;\ u'=a'b+b'a;$
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$a=x;\ b=\ln(x+\sqrt{1+x^2})$
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$a'=1;$
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$b=\ln t;\ b'=\dfrac{t'}{t};$
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$t=x+\sqrt{1+x^2};$
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$t'=1+(\sqrt{1+x^2})';$
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$(\sqrt{1+x^2})'=\dfrac{1}{2}(1+x^2)^{-\frac{1}{2}}2x=\dfrac{x}{\sqrt{1+x^2}};$
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$t'=1+\dfrac{x}{\sqrt{1+x^2}}=\dfrac{\sqrt{1+x^2}+x}{\sqrt{1+x^2}};$
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$b'=\dfrac{\frac{\sqrt{1+x^2}+x}{\sqrt{1+x^2}}}{x+\sqrt{1+x^2}}=\dfrac{1}{\sqrt{1+x^2}};$
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$u'=\ln(x+\sqrt{1+x^2})+x\cdot\dfrac{1}{\sqrt{1+x^2}};$
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$v=(1+x^2)^{\frac{1}{2}};\ v'=\dfrac{1}{2}(1+x^2)^{-\frac{1}{2}}2x=\dfrac{x}{\sqrt{1+x^2}};$
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$y'=(\ln(x+\sqrt{1+x^2})+x\cdot\dfrac{1}{\sqrt{1+x^2}})-\dfrac{x}{\sqrt{1+x^2}}=\ln(x+\sqrt{1+x^2})$
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## 42 номер - Д 900
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### Пример:
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$y=\dfrac{2+3x^{2}}{x^{4}}\sqrt{ 1-x^{2} }+3\ln \dfrac{1+\sqrt{ 1-x^{2} }}{x}$
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$y'=?$
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### Решение:
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$\dfrac{2+3x^{2}}{x^{4}}\sqrt{1-x^{2}}+3\ln\dfrac{1+\sqrt{1-x^{2}}}{x}$
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$y=u+v;\ y'=u'+v';$
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$u=\dfrac{2+3x^{2}}{x^{4}}\sqrt{1-x^{2}};\ v=3\ln\dfrac{1+\sqrt{1-x^{2}}}{x}$
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$u=ab;\ u'=a'b+ab';$
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$a=\dfrac{2+3x^{2}}{x^{4}};\ b=\sqrt{1-x^{2}}$
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$a=(2+3x^{2})x^{-4};$
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$a'=6x\cdot x^{-4}+(2+3x^{2})(-4)x^{-5}=\dfrac{6}{x^{3}}-\dfrac{8+12x^{2}}{x^{5}}=-\dfrac{6x^{2}+8}{x^{5}};$
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$b=(1-x^{2})^{\frac{1}{2}};\ b'=\dfrac{1}{2}(1-x^{2})^{-\frac{1}{2}}(-2x)=-\dfrac{x}{\sqrt{1-x^{2}}};$
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$u'=-\dfrac{6x^{2}+8}{x^{5}}\sqrt{1-x^{2}}+\dfrac{2+3x^{2}}{x^{4}}(-\dfrac{x}{\sqrt{1-x^{2}}})=-\dfrac{6x^{2}+8}{x^{5}}\sqrt{1-x^{2}}-\dfrac{2+3x^{2}}{x^{3}\sqrt{1-x^{2}}}$
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$v=3(\ln(1+\sqrt{1-x^{2}})-\ln x);$
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$v'=3(\dfrac{(\sqrt{1-x^{2}})'}{1+\sqrt{1-x^{2}}}-\dfrac{1}{x});$
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$(\sqrt{1-x^{2}})'=-\dfrac{x}{\sqrt{1-x^{2}}};$
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$v'=3(-\dfrac{x}{\sqrt{1-x^{2}}(1+\sqrt{1-x^{2}})}-\dfrac{1}{x})=-\dfrac{3}{x\sqrt{1-x^{2}}}$
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$y'=-\dfrac{6x^{2}+8}{x^{5}}\sqrt{1-x^{2}}-\dfrac{2+6x^{2}}{x^{3}\sqrt{1-x^{2}}}=-\dfrac{6x^{2}+8}{x^{5}}\cdot\dfrac{1-x^{2}}{\sqrt{1-x^{2}}}-\dfrac{(2+6x^{2})x^{2}}{x^{5}\sqrt{1-x^{2}}}=-\dfrac{(6x^{2}+8)(1-x^{2})+(2+6x^{2})x^{2}}{x^{5}\sqrt{1-x^{2}}}=-\dfrac{8}{x^{5}\sqrt{1-x^{2}}}$
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## 46 номер - Д 904
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### Пример:
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$y=\ln \sqrt{ \dfrac{1-\sin x}{1+\sin x} }$
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$y'=?$
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### Решение:
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$\ln\sqrt{\dfrac{1-\sin x}{1+\sin x}}$
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$y=\ln((\dfrac{1-\sin x}{1+\sin x})^{\frac{1}{2}})=\dfrac{1}{2}\ln\dfrac{1-\sin x}{1+\sin x}=\dfrac{1}{2}(\ln(1-\sin x)-\ln(1+\sin x))$
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$y'=\dfrac{1}{2}(\dfrac{-(\sin x)'}{1-\sin x}-\dfrac{(\sin x)'}{1+\sin x})=\dfrac{1}{2}(-\dfrac{\cos x}{1-\sin x}-\dfrac{\cos x}{1+\sin x})$
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$y'=-\dfrac{\cos x}{2}\cdot\dfrac{(1+\sin x)+(1-\sin x)}{1-\sin^{2}x}=-\dfrac{1}{\cos x}$
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## 50 номер - Д 908
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### Пример:
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$y= \dfrac{1}{4x^{4}}\ln \dfrac{1}{x}- \dfrac{1}{16x^{4}}$
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$y'=?$
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### Решение:
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$\dfrac{1}{4x^{4}}\ln\dfrac{1}{x}-\dfrac{1}{16x^{4}}$
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$y=u+v;\ y'=u'+v';$
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$u=\dfrac{1}{4x^{4}}\ln\dfrac{1}{x};\ v=-\dfrac{1}{16x^{4}}$
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$u=\dfrac{1}{4}ab;\ u'=\dfrac{1}{4}(a'b+ab');$
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$a=x^{-4};\ b=\ln\dfrac{1}{x}$
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$a'=-4x^{-5}=-\dfrac{4}{x^{5}};$
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$b=\ln(x^{-1});\ b'=-(\ln x)'=-\dfrac{1}{x};$
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$u'=\dfrac{1}{4}(-\dfrac{4}{x^{5}}\ln\dfrac{1}{x}+x^{-4}(-\dfrac{1}{x}))=-\dfrac{1}{x^{5}}\ln\dfrac{1}{x}-\dfrac{1}{4x^{5}}$
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$v'=-\dfrac{1}{16}(-4)x^{-5}=\dfrac{1}{4x^{5}}$
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$y'=-\dfrac{1}{x^{5}}\ln\dfrac{1}{x}$
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## 53 номер - Д 963
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### Пример:
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$y=\sqrt[ x ]{ x }; (x>0)$
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$y'=?$
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### Решение:
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$y=\sqrt[x]{x}=x^{\frac{1}{x}}$
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$\ln y=\ln(x^{\frac{1}{x}})=\dfrac{1}{x}\ln x$
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$\dfrac{y'}{y}=(\dfrac{\ln x}{x})'$
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$u=\ln x;\ v=x;\ (\dfrac{u}{v})'=\dfrac{u'v-uv'}{v^{2}};$
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$u'=\dfrac{1}{x};\ v'=1;$
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$(\dfrac{\ln x}{x})'=\dfrac{\dfrac{1}{x}\cdot x-\ln x}{x^{2}}=\dfrac{1-\ln x}{x^{2}}$
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$y'=y\cdot\dfrac{1-\ln x}{x^{2}}=x^{\frac{1}{x}}\cdot\dfrac{1-\ln x}{x^{2}}$
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## 54 номер - Д 964
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### Пример:
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$y=(\sin x)^{\cos x}+(\cos x)^{\sin x}$
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$y'=?$
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### Решение:
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$(\sin x)^{\cos x}+(\cos x)^{\sin x}$
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$y=u+v;\ y'=u'+v';$
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$u=(\sin x)^{\cos x};\ v=(\cos x)^{\sin x}$
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$\ln u=\cos x\ln(\sin x)$
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$\dfrac{u'}{u}=(\cos x)'\ln(\sin x)+\cos x(\ln(\sin x))'=-\sin x\ln(\sin x)+\cos x\cdot\dfrac{\cos x}{\sin x}$
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$u'=u(-\sin x\ln(\sin x)+\dfrac{\cos^{2}x}{\sin x})=(\sin x)^{\cos x}(-\sin x\ln(\sin x)+\dfrac{\cos^{2}x}{\sin x})$
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$\ln v=\sin x\ln(\cos x)$
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$\dfrac{v'}{v}=(\sin x)'\ln(\cos x)+\sin x(\ln(\cos x))'=\cos x\ln(\cos x)+\sin x(-\dfrac{\sin x}{\cos x})$
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$v'=v(\cos x\ln(\cos x)-\dfrac{\sin^{2}x}{\cos x})=(\cos x)^{\sin x}(\cos x\ln(\cos x)-\dfrac{\sin^{2}x}{\cos x})$
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$y'=(\sin x)^{\cos x}(-\sin x\ln(\sin x)+\dfrac{\cos^{2}x}{\sin x})+(\cos x)^{\sin x}(\cos x\ln(\cos x)-\dfrac{\sin^{2}x}{\cos x})$
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## 57 номер - Д 984Б
|
||
|
||
### Пример:
|
||
|
||
$y=\dfrac{x^{2}}{1-x}\sqrt[ 3 ]{ \dfrac{3-x}{(3+x)^{2}} }$
|
||
$y'=?$
|
||
|
||
### Решение:
|
||
|
||
$y=\dfrac{x^{2}}{1-x}\sqrt[3]{\dfrac{3-x}{(3+x)^{2}}}$
|
||
|
||
$\ln y=\ln(\dfrac{x^{2}}{1-x})+\ln((\dfrac{3-x}{(3+x)^{2}})^{\frac{1}{3}})=(2\ln x-\ln(1-x))+\dfrac{1}{3}(\ln(3-x)-2\ln(3+x))$
|
||
|
||
$\dfrac{y'}{y}=(2\ln x-\ln(1-x))'+\dfrac{1}{3}(\ln(3-x)-2\ln(3+x))'$
|
||
$\dfrac{y'}{y}=\dfrac{2}{x}-(\ln(1-x))'+\dfrac{1}{3}(\dfrac{-1}{3-x}-2\cdot\dfrac{1}{3+x})$
|
||
$(\ln(1-x))'=\dfrac{(1-x)'}{1-x}=-\dfrac{1}{1-x}$
|
||
$\dfrac{y'}{y}=\dfrac{2}{x}+\dfrac{1}{1-x}-\dfrac{1}{3}(\dfrac{1}{3-x}+\dfrac{2}{3+x})$
|
||
|
||
$y'=y(\dfrac{2}{x}+\dfrac{1}{1-x}-\dfrac{1}{3}(\dfrac{1}{3-x}+\dfrac{2}{3+x}))$
|
||
|
||
## 58 номер - Д 984В
|
||
|
||
### Пример:
|
||
|
||
$y=(x-a_{1})^{a_{1}}(x-a_{2})^{a_{2}}\dots(x-a_{n})^{a_{n}}$
|
||
$y'=?$
|
||
|
||
### Решение:
|
||
|
||
$y=(x-a_{1})^{a_{1}}(x-a_{2})^{a_{2}}\dots(x-a_{n})^{a_{n}}$
|
||
|
||
$\ln y=\ln((x-a_{1})^{a_{1}}(x-a_{2})^{a_{2}}\dots(x-a_{n})^{a_{n}})=\ln(x-a_{1})^{a_{1}}+\ln(x-a_{2})^{a_{2}}+\dots+\ln(x-a_{n})^{a_{n}}$
|
||
$\ln y=a_{1}\ln(x-a_{1})+a_{2}\ln(x-a_{2})+\dots+a_{n}\ln(x-a_{n})$
|
||
|
||
$\dfrac{y'}{y}=(a_{1}\ln(x-a_{1})+a_{2}\ln(x-a_{2})+\dots+a_{n}\ln(x-a_{n}))'$
|
||
$\dfrac{y'}{y}=\dfrac{a_{1}}{x-a_{1}}+\dfrac{a_{2}}{x-a_{2}}+\dots+\dfrac{a_{n}}{x-a_{n}}$
|
||
|
||
$y'=y(\dfrac{a_{1}}{x-a_{1}}+\dfrac{a_{2}}{x-a_{2}}+\dots+\dfrac{a_{n}}{x-a_{n}})$
|
||
$y'=(x-a_{1})^{a_{1}}(x-a_{2})^{a_{2}}\dots(x-a_{n})^{a_{n}}(\dfrac{a_{1}}{x-a_{1}}+\dfrac{a_{2}}{x-a_{2}}+\dots+\dfrac{a_{n}}{x-a_{n}})$
|
||
|
||
## 61 номер - Д 985Б
|
||
|
||
### Пример:
|
||
|
||
$y=\text{arccot} \dfrac{\phi(x)}{\psi(x)}$
|
||
$y'=?$
|
||
|
||
### Решение:
|
||
|
||
$y=\text{arccot}\dfrac{\phi(x)}{\psi(x)}$
|
||
|
||
$y=\text{arccot}(u);\ y'=-\dfrac{u'}{1+u^{2}};$
|
||
$u=\dfrac{\phi(x)}{\psi(x)}$
|
||
|
||
$u=\dfrac{p}{q};\ u'=\dfrac{p'q-pq'}{q^{2}};$
|
||
$p=\phi(x);\ q=\psi(x)$
|
||
|
||
$u'=\dfrac{\phi'(x)\psi(x)-\phi(x)\psi'(x)}{\psi^{2}(x)}$
|
||
$1+u^{2}=1+\dfrac{\phi^{2}(x)}{\psi^{2}(x)}=\dfrac{\psi^{2}(x)+\phi^{2}(x)}{\psi^{2}(x)}$
|
||
|
||
$y'=-\dfrac{\dfrac{\phi'\psi-\phi\psi'}{\psi^{2}}}{\dfrac{\psi^{2}+\phi^{2}}{\psi^{2}}}=-\dfrac{\phi'\psi-\phi\psi'}{\phi^{2}+\psi^{2}}=\dfrac{\phi\psi'-\phi'\psi}{\phi^{2}+\psi^{2}}$
|
||
|
||
## 65 номер - Д 989
|
||
|
||
### Пример:
|
||
|
||
$F(x)=\begin{vmatrix}x & x^{2} & x^{3} \\ 1 & 2x & 3x^{2} \\ 0 & 2 & 6x\end{vmatrix}$
|
||
$F(x)'=?$
|
||
|
||
### Решение:
|
||
|
||
$F(x)=\begin{vmatrix}x & x^{2} & x^{3} \\ 1 & 2x & 3x^{2} \\ 0 & 2 & 6x\end{vmatrix}$
|
||
|
||
$F(x)=x\begin{vmatrix}2x & 3x^{2} \\ 2 & 6x\end{vmatrix}-x^{2}\begin{vmatrix}1 & 3x^{2} \\ 0 & 6x\end{vmatrix}+x^{3}\begin{vmatrix}1 & 2x \\ 0 & 2\end{vmatrix}$
|
||
$F(x)=x(2x\cdot6x-3x^{2}\cdot2)-x^{2}(1\cdot6x-0)+x^{3}(1\cdot2-0)$
|
||
$F(x)=x(12x^{2}-6x^{2})-6x^{3}+2x^{3}=6x^{3}-6x^{3}+2x^{3}=2x^{3}$
|
||
|
||
$F'(x)=6x^{2}$
|
||
|
||
## 69 номер - Д 1042
|
||
|
||
### Пример:
|
||
|
||
Найти производные $y'_{x}$ (параметры положительны)
|
||
$x=a\cosh t$
|
||
$y=b \sinh t$
|
||
$y'_{x}=?$
|
||
|
||
### Решение:
|
||
|
||
$x=a\cosh t;\ y=b\sinh t$
|
||
|
||
$\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$
|
||
$\dfrac{dx}{dt}=a(\cosh t)'=a\sinh t$
|
||
$\dfrac{dy}{dt}=b(\sinh t)'=b\cosh t$
|
||
|
||
$y'_{x}=\dfrac{b\cosh t}{a\sinh t}=\dfrac{b}{a}\cdot\dfrac{\cosh t}{\sinh t}=\dfrac{b}{a}\text{cth}\ t$
|
||
|
||
## 73 номер - Д 1050
|
||
|
||
### Пример:
|
||
|
||
$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1 \ (эллипс)$
|
||
$y'=?$
|
||
|
||
### Решение:
|
||
|
||
$\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$
|
||
|
||
$(\dfrac{x^{2}}{a^{2}})'+(\dfrac{y^{2}}{b^{2}})'=0$
|
||
$\dfrac{2x}{a^{2}}+\dfrac{2y}{b^{2}}y'=0$
|
||
$y'=-\dfrac{2x}{a^{2}}\cdot\dfrac{b^{2}}{2y}=-\dfrac{b^{2}x}{a^{2}y}$
|
||
|
||
## 77 номер - Д 1086
|
||
|
||
### Пример:
|
||
|
||
$y=\dfrac{1}{a}\text{arccot} \dfrac{x}{a}; (a\neq0)$
|
||
|
||
### Решение:
|
||
|
||
$a=\text{const};$
|
||
$y=\dfrac{1}{a}\text{arccot} u;\ dy=\dfrac{1}{a}d(\text{arccot} u);$
|
||
$d(\text{arccot} u)=-\dfrac{1}{1+u^{2}}du;$
|
||
$u=\dfrac{x}{a};\ du=\dfrac{1}{a}dx;$
|
||
|
||
$dy=\dfrac{1}{a}(-\dfrac{1}{1+(\frac{x}{a})^{2}}\cdot\dfrac{1}{a}dx)=-\dfrac{1}{a^{2}}\cdot\dfrac{1}{1+\frac{x^{2}}{a^{2}}}dx=-\dfrac{1}{a^{2}}\cdot\dfrac{a^{2}}{a^{2}+x^{2}}dx=-\dfrac{dx}{a^{2}+x^{2}}$
|
||
|
||
## 80 номер - Д 1088
|
||
|
||
### Пример:
|
||
|
||
$y=\ln|x+\sqrt{ x^{2+a} }|$
|
||
|
||
### Решение:
|
||
|
||
$a=\text{const};$
|
||
$y=\ln|u|;\ dy=\dfrac{du}{u};$
|
||
$u=x+\sqrt{x^{2+a}}$
|
||
|
||
$du=dx+d(\sqrt{x^{2+a}});$
|
||
$\sqrt{x^{2+a}}=(x^{2+a})^{\frac{1}{2}};$
|
||
$d((x^{2+a})^{\frac{1}{2}})=\dfrac{1}{2}(x^{2+a})^{-\frac{1}{2}}d(x^{2+a});$
|
||
$d(x^{2+a})=(2+a)x^{1+a}dx;$
|
||
$d(\sqrt{x^{2+a}})=\dfrac{1}{2}(x^{2+a})^{-\frac{1}{2}}(2+a)x^{1+a}dx=\dfrac{2+a}{2}x^{\frac{a}{2}}dx;$
|
||
|
||
$du=(1+\dfrac{2+a}{2}x^{\frac{a}{2}})dx;$
|
||
$dy=\dfrac{(1+\dfrac{2+a}{2}x^{\frac{a}{2}})dx}{x+\sqrt{x^{2+a}}}$
|
||
|
||
## 81 номер - Д 1089
|
||
|
||
### Пример:
|
||
|
||
$y=\arcsin \dfrac{x}{a}; (a\neq 0)$
|
||
|
||
### Решение:
|
||
|
||
$a=\text{const};$
|
||
$y=\arcsin u;\ dy=d(\arcsin u);$
|
||
$d(\arcsin u)=\dfrac{1}{\sqrt{1-u^{2}}}du;$
|
||
$u=\dfrac{x}{a};\ du=\dfrac{1}{a}dx;$
|
||
|
||
$dy=\dfrac{1}{\sqrt{1-(\frac{x}{a})^{2}}}\cdot\dfrac{1}{a}dx=\dfrac{dx}{a\sqrt{1-(\frac{x}{a})^{2}}}$
|
||
|
||
## 84 номер - Д 1090В
|
||
|
||
### Пример:
|
||
|
||
$d(\dfrac{1}{x^{3}})$
|
||
|
||
### Решение:
|
||
|
||
$\dfrac{1}{x^3}=x^{-3}$
|
||
$d(x^{-3})=(-3)x^{-4}dx=-\dfrac{3}{x^{4}}dx$
|
||
|
||
## 85 номер - Д 1090Г
|
||
|
||
### Пример:
|
||
|
||
$d(\dfrac{\ln x}{\sqrt{ x }})$
|
||
|
||
### Решение:
|
||
|
||
$\dfrac{\ln x}{\sqrt{x}}=\ln x\cdot x^{-\frac{1}{2}}$
|
||
|
||
$d(uv)=u\,dv+v\,du;$
|
||
$u=\ln x;\ v=x^{-\frac{1}{2}}$
|
||
|
||
$du=\dfrac{1}{x}dx;$
|
||
$dv=-\dfrac{1}{2}x^{-\frac{3}{2}}dx;$
|
||
|
||
$d(\dfrac{\ln x}{\sqrt{x}})=\ln x(-\dfrac{1}{2}x^{-\frac{3}{2}}dx)+x^{-\frac{1}{2}}(\dfrac{1}{x}dx)=(-\dfrac{\ln x}{2x^{\frac{3}{2}}}+\dfrac{1}{x^{\frac{3}{2}}})dx=\dfrac{2-\ln x}{2x^{\frac{3}{2}}}dx$
|
||
|
||
## 88 номер - Д 1093
|
||
|
||
### Пример:
|
||
|
||
$y=\dfrac{1}{\sqrt{ u^{2}+v^{2} }}$
|
||
|
||
### Решение:
|
||
|
||
$y=(u^2+v^2)^{-\frac{1}{2}}$
|
||
$dy=-\dfrac{1}{2}(u^2+v^2)^{-\frac{3}{2}}d(u^2+v^2)$
|
||
|
||
$d(u^2+v^2)=d(u^2)+d(v^2)=2u\,du+2v\,dv;$
|
||
$dy=-\dfrac{1}{2}(u^2+v^2)^{-\frac{3}{2}}(2u\,du+2v\,dv)=-\dfrac{u\,du+v\,dv}{(u^2+v^2)^{\frac{3}{2}}}$
|
||
|
||
## 89 номер - Д 1094
|
||
|
||
### Пример:
|
||
|
||
$y=\text{arccon} \dfrac{u}{v}$
|
||
|
||
### Решение:
|
||
|
||
$y=\text{arccon}\,w;\ dy=d(\text{arccon}\,w);$
|
||
$d(\text{arccon}\,w)=-\dfrac{1}{\sqrt{1-w^2}}dw;$
|
||
$w=\dfrac{u}{v}$
|
||
|
||
$dw=d(\dfrac{u}{v})=\dfrac{v\,du-u\,dv}{v^2};$
|
||
$dy=-\dfrac{1}{\sqrt{1-(\frac{u}{v})^2}}\cdot\dfrac{v\,du-u\,dv}{v^2}$
|
||
|
||
## 92 номер - Д 1100
|
||
|
||
### Пример:
|
||
|
||
$\sin 29\degree\approx \ ?$
|
||
|
||
### Решение:
|
||
|
||
$y=\sin x;$
|
||
$x_0=30\degree=\dfrac{\pi}{6};$
|
||
$\Delta x=29\degree-30\degree=-1\degree=-\dfrac{\pi}{180};$
|
||
$y(x_0+\Delta x)\approx y(x_0)+y'(x_0)\Delta x;$
|
||
$y'=\cos x;$
|
||
|
||
$\sin29\degree\approx\sin30\degree+\cos30\degree(-\dfrac{\pi}{180})=\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}\cdot\dfrac{\pi}{180}=\dfrac{1}{2}-\dfrac{\sqrt{3}\pi}{360}\approx0,485;$
|
||
|
||
## 96 номер - Д 1103
|
||
|
||
### Пример:
|
||
|
||
$lg 11 \approx \ ?$
|
||
|
||
### Решение:
|
||
|
||
$y=lgx;$
|
||
$x_0=10;\ \Delta x=1;$
|
||
$y(x_0+\Delta x)\approx y(x_0)+y'(x_0)\Delta x;$
|
||
$(lgx)'=\dfrac{1}{x\ln10};$
|
||
|
||
$lg11\approx lg10+\dfrac{1}{10\ln10}\cdot1=1+\dfrac{1}{10\ln10}\approx1,043;$
|
||
|
||
## 100 номер - Д 1105А
|
||
|
||
### Пример:
|
||
|
||
$\sqrt[ 3 ]{ 9 }\approx \ ?$
|
||
|
||
### Решение:
|
||
|
||
$\sqrt[ 3 ]{ 9 }=\sqrt[3]{8+1};$
|
||
$n=3;\ a=2;\ x=1;\ (a>0)$
|
||
$\sqrt[n]{a^{n}+x}\approx a+\dfrac{x}{na^{n-1}}$
|
||
|
||
$\sqrt[3]{9}\approx2+\dfrac{1}{3\cdot2^{2}}=2+\dfrac{1}{12}\approx2,083;$
|
||
|
||
## 104 номер - РИСУНОК
|
||
![[telegram-cloud-document-2-5407087289899717974.jpg]]
|
||
|
||
## 105 номер - АНЕКДОТ
|
||
На одном корабле работал фокусник. Так как пассажиры постоянно менялись, он без перемены проделывал одни и те же фокусы. К его несчастью, капитанский попугай просмотрел его выступления достаточно раз, чтобы разгадать все секреты. Во время каждого выступления попугай портил все фокусы своими криками «Эта не та шляпа! Он прячет пиковую даму в кармане брюк! В коробке дырочка!». Фокусник сердился, но ничего поделать не мог, попугай всё-таки капитанский.
|
||
Однажды корабль потерпел кораблекрушение, и только фокусник с попугаем чудом выжили. Продолжали они плавать в море на каком-то бревне. Фокусник постоянно злобно смотрел на попугая, который в свою очередь не переставал смотреть на фокусника. Наконец, через неделю дрейфа попугай не выдержал:
|
||
\- Ну ладно, ладно, сдаюсь! Куда ты корабль засунул то?!
|
||
|
||
|
||
## 108 номер - Д 1133
|
||
|
||
### Пример:
|
||
|
||
$y=x^{x}$
|
||
$d^{2}y=?$
|
||
|
||
### Решение:
|
||
|
||
$x=\text{независимая};\ d(dx)=0;$
|
||
$y=x^x$
|
||
|
||
$\ln y=x\ln x$
|
||
$\dfrac{dy}{y}=d(x\ln x)=(x\ln x)'dx=(\ln x+1)dx$
|
||
$dy=y(\ln x+1)dx$
|
||
|
||
$d^{2}y=d(dy)=d(y(\ln x+1)dx)=d(y(\ln x+1))dx$
|
||
$d(y(\ln x+1))=(\ln x+1)dy+y\,d(\ln x+1)$
|
||
$d(\ln x+1)=\dfrac{1}{x}dx$
|
||
$d(y(\ln x+1))=(\ln x+1)\,y(\ln x+1)dx+y\cdot\dfrac{1}{x}dx=y((\ln x+1)^{2}+\dfrac{1}{x})dx$
|
||
|
||
$d^{2}y=x^{x}((\ln x+1)^{2}+\dfrac{1}{x})dx^{2}$
|
||
|
||
## 111 номер - Д 1142
|
||
|
||
### Пример:
|
||
|
||
$x=a(t-\sin t)$
|
||
$y=a(1-\cos t)$
|
||
$y'''=?$
|
||
|
||
### Решение:
|
||
|
||
$x=a(t-\sin t);\ y=a(1-\cos t)$
|
||
|
||
$\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$
|
||
$\dfrac{dx}{dt}=a(1-\cos t)$
|
||
$\dfrac{dy}{dt}=a\sin t$
|
||
$y'_{x}=\dfrac{a\sin t}{a(1-\cos t)}=\dfrac{\sin t}{1-\cos t}$
|
||
|
||
$y''_{x}=\dfrac{\dfrac{d}{dt}(y'_{x})}{\dfrac{dx}{dt}}$
|
||
$\dfrac{d}{dt}(\dfrac{\sin t}{1-\cos t})=\dfrac{(\cos t)(1-\cos t)-\sin t\cdot\sin t}{(1-\cos t)^{2}}=\dfrac{\cos t-1}{(1-\cos t)^{2}}=-\dfrac{1}{1-\cos t}$
|
||
$y''_{x}=\dfrac{-\dfrac{1}{1-\cos t}}{a(1-\cos t)}=-\dfrac{1}{a(1-\cos t)^{2}}$
|
||
|
||
$y'''_{x}=\dfrac{\dfrac{d}{dt}(y''_{x})}{\dfrac{dx}{dt}}$
|
||
$\dfrac{d}{dt}(-\dfrac{1}{a}(1-\cos t)^{-2})=-\dfrac{1}{a}(-2)(1-\cos t)^{-3}\sin t=\dfrac{2\sin t}{a(1-\cos t)^{3}}$
|
||
$y'''_{x}=\dfrac{\dfrac{2\sin t}{a(1-\cos t)^{3}}}{a(1-\cos t)}=\dfrac{2\sin t}{a^{2}(1-\cos t)^{4}}$
|
||
|
||
## 112 номер - Д 1143
|
||
|
||
### Пример:
|
||
|
||
$x=e^{ t }\cos t$
|
||
$y=e^{ t }\sin t$
|
||
$y'''=?$
|
||
|
||
### Решение:
|
||
|
||
$x=e^{t}\cos t;\ y=e^{t}\sin t$
|
||
|
||
$\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}$
|
||
$\dfrac{dx}{dt}=e^{t}\cos t+e^{t}(-\sin t)=e^{t}(\cos t-\sin t)$
|
||
$\dfrac{dy}{dt}=e^{t}\sin t+e^{t}\cos t=e^{t}(\sin t+\cos t)$
|
||
$y'_{x}=\dfrac{e^{t}(\sin t+\cos t)}{e^{t}(\cos t-\sin t)}=\dfrac{\sin t+\cos t}{\cos t-\sin t}$
|
||
|
||
$y''_{x}=\dfrac{\dfrac{d}{dt}(y'_{x})}{\dfrac{dx}{dt}}$
|
||
$\dfrac{d}{dt}(\dfrac{\sin t+\cos t}{\cos t-\sin t})=\dfrac{(\cos t-\sin t)(\cos t-\sin t)-(\sin t+\cos t)(-\sin t-\cos t)}{(\cos t-\sin t)^{2}}$
|
||
$\dfrac{d}{dt}(\dfrac{\sin t+\cos t}{\cos t-\sin t})=\dfrac{(\cos t-\sin t)^{2}+(\sin t+\cos t)^{2}}{(\cos t-\sin t)^{2}}=\dfrac{2}{(\cos t-\sin t)^{2}}$
|
||
$y''_{x}=\dfrac{\dfrac{2}{(\cos t-\sin t)^{2}}}{e^{t}(\cos t-\sin t)}=\dfrac{2}{e^{t}(\cos t-\sin t)^{3}}$
|
||
|
||
$y'''_{x}=\dfrac{\dfrac{d}{dt}(y''_{x})}{\dfrac{dx}{dt}}$
|
||
$y''_{x}=2e^{-t}(\cos t-\sin t)^{-3}$
|
||
$\dfrac{d}{dt}(y''_{x})=2((-e^{-t})(\cos t-\sin t)^{-3}+e^{-t}(-3)(\cos t-\sin t)^{-4}(-\sin t-\cos t))$
|
||
$\dfrac{d}{dt}(y''_{x})=2e^{-t}(-(\cos t-\sin t)^{-3}+3(\sin t+\cos t)(\cos t-\sin t)^{-4})$
|
||
$\dfrac{d}{dt}(y''_{x})=\dfrac{2e^{-t}(-(\cos t-\sin t)+3(\sin t+\cos t))}{(\cos t-\sin t)^{4}}=\dfrac{4e^{-t}(\cos t+2\sin t)}{(\cos t-\sin t)^{4}}$
|
||
$y'''_{x}=\dfrac{\dfrac{4e^{-t}(\cos t+2\sin t)}{(\cos t-\sin t)^{4}}}{e^{t}(\cos t-\sin t)}=\dfrac{4(\cos t+2\sin t)}{e^{2t}(\cos t-\sin t)^{5}}$
|
||
|
||
## 115 номер - Д 1157
|
||
|
||
### Пример:
|
||
|
||
$y= \dfrac{a}{x^{m}}$
|
||
$y'''=?$
|
||
|
||
### Решение:
|
||
|
||
$y=a x^{-m};\ a,m=\text{const};$
|
||
|
||
$y'=a(-m)x^{-m-1}$
|
||
$y''=a(-m)(-m-1)x^{-m-2}$
|
||
$y'''=a(-m)(-m-1)(-m-2)x^{-m-3}=-\dfrac{am(m+1)(m+2)}{x^{m+3}}$
|
||
|
||
## 116 номер - Д 1159
|
||
|
||
### Пример:
|
||
|
||
$y=\dfrac{x^{2}}{1-x}$
|
||
$y^{(8)}=?$
|
||
|
||
### Решение:
|
||
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$y=\dfrac{x^{2}}{1-x}=-x-1+\dfrac{1}{1-x}$
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$\dfrac{1}{1-x}=(1-x)^{-1}$
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|
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$((1-x)^{-1})'=(1-x)^{-2}$
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$((1-x)^{-2})'=2(1-x)^{-3}$
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$((1-x)^{-3})'=3\cdot2(1-x)^{-4}$
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|
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$((1-x)^{-1})^{(n)}=n!(1-x)^{-(n+1)};\ (n\ge1)$
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$(-x-1)^{(8)}=0$
|
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$y^{(8)}=\dfrac{8!}{(1-x)^{9}}$
|
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## 119 номер - Д 1163
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### Пример:
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|
||
$y=x\ln x$
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$y^{(5)}=?$
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|
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### Решение:
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||
|
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$y=x\ln x$
|
||
|
||
$y'=(x\ln x)'=\ln x+1$
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||
$y''=(\ln x+1)'=\dfrac{1}{x}$
|
||
$y'''=(\dfrac{1}{x})'=-\dfrac{1}{x^{2}}$
|
||
$y^{(4)}=(-\dfrac{1}{x^{2}})'=\dfrac{2}{x^{3}}$
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$y^{(5)}=(\dfrac{2}{x^{3}})'=-\dfrac{6}{x^{4}}$
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