69 lines
1.4 KiB
Markdown
69 lines
1.4 KiB
Markdown
#учеба #семестр_1 #высшая_математика
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1. $y=5x^{4}$
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$y'=20x^{3}$
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2. $y=\sqrt{ x }$
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$y'=\dfrac{1}{2\sqrt{ x }}$
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3. $y=\dfrac{1}{x^{3}}$
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$y'=-3 \dfrac{1}{x^{4}}$
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4. $y=3x^{2}-7x+1$
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$y'=6x-7$
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5. $y=(x-2)^{5}$
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$a=(x-2)^{5}$
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$a'=5(x-2)^{4}$
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$b=x-2$
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$b'=1$
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$y'=a'b\cdot b'=5(x-2)^{4}$
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6. $y=x^{2}\sin x$
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$y'=ab'+a'b$
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$a=x^{2}$
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$a'=2x$
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$b=\sin x$
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$b'=\cos x$
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$y'=x^{2}\cos x+2x\sin x$
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7. $y=(x+1)e^{ x }$
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$y'=ab'+a'b$
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$a=(x+1)$
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$a'=1$
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$b=b'=e^{ x }$
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$y'=e^{ x }+(x+1)e^{ x }=e^{ x }(x+2)$
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8. $y=\dfrac{x^{2}+1}{x-1}$
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$y'=\dfrac{a'b-ab'}{b^{2}}$
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$a=x^{2}+1$
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$a'=2x$
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$b=x-1$
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$b'=1$
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$y'=\dfrac{2x(x-1)-(x^{2}+1)}{(x-1)^{2}}$
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9. $y=\dfrac{\ln x}{x}$
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$y'=\dfrac{a'b-ab'}{b^{2}}$
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$a=\ln x$
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$a'=\dfrac{1}{x}$
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$b=x$
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$b'=1$
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$y'=\dfrac{1-\ln x}{x^{2}}$
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10. $y=\sin(3x)$
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$y'=3\cos(3x)$
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11. $y=\cos(x^{2})$
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$y'=-2x\sin(x^{2})$
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12. $y=e^{ 2x-1 }$
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$y'=2e^{ 2x-1 }$
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13. $y=\ln(5x+2)$
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$y'=\dfrac{5}{5x+2}$
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14. $y=\sqrt{ 1-x^{2} }$
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$y'=-\dfrac{1}{\sqrt{ 1-x^{2} }}$
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15. $y=\dfrac{x}{(1-x)^{2}(1+x)^{3}}$
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$y'=\dfrac{a'b-ab'}{b^{2}}$
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$a=x$
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$a'=1$
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$b=(1-x)^{2}(1+x)^{3}$
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$b'=mn'+m'n$
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$m=(1-x)^{2}$
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$m'=-2(1-x)$
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$n=(1+x)^{3}$
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$n'=3(1+x)^{2}$
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$b'=(1-x)^{2}\cdot3(1+x)^{2}-2(1-x)\cdot(1+x)^{3}=(1-x)(1+x)^{2}(3(1-x)-2(1+x))$
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$y'=\dfrac{(1-x)^{2}(1+x)^{3}-x((1-x)(1+x)^{2}(3(1-x)-2(1+x)))}{((1-x)^{2}(1+x)^{3})^{2}}$
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