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Производная сложной функции:
![f(g(x))' = f'(g(x))*g'(x) f(g(x))' = f'(g(x))*g'(x)](https://tex.z-dn.net/?f=f%28g%28x%29%29%27+%3D+f%27%28g%28x%29%29%2Ag%27%28x%29)
1.
![(sin8x)'=cos(8x)*(8x)'=cos(8x) * 8=8cos(8x) (sin8x)'=cos(8x)*(8x)'=cos(8x) * 8=8cos(8x)](https://tex.z-dn.net/?f=%28sin8x%29%27%3Dcos%288x%29%2A%288x%29%27%3Dcos%288x%29+%2A+8%3D8cos%288x%29)
2.![( sqrt{ x^{2} +3x} )' =((x^{2} +3x)^{ frac{1}{2} }) = frac{1}{2}*(x^{2} +3x )^{- frac{1}{2}} *(x^{2} +3x)' = ( sqrt{ x^{2} +3x} )' =((x^{2} +3x)^{ frac{1}{2} }) = frac{1}{2}*(x^{2} +3x )^{- frac{1}{2}} *(x^{2} +3x)' =](https://tex.z-dn.net/?f=%28+sqrt%7B+x%5E%7B2%7D+%2B3x%7D+%29%27+%3D%28%28x%5E%7B2%7D+%2B3x%29%5E%7B+frac%7B1%7D%7B2%7D+%7D%29+%3D+frac%7B1%7D%7B2%7D%2A%28x%5E%7B2%7D+%2B3x+%29%5E%7B-+frac%7B1%7D%7B2%7D%7D+%2A%28x%5E%7B2%7D+%2B3x%29%27+%3D+)
![=frac{1}{2}* frac{1}{ (x^{2} +3x )^{frac{1}{2}} } * (2x+3)= frac{2x+3}{2 sqrt{x^{2} +3x} } =frac{1}{2}* frac{1}{ (x^{2} +3x )^{frac{1}{2}} } * (2x+3)= frac{2x+3}{2 sqrt{x^{2} +3x} }](https://tex.z-dn.net/?f=+%3Dfrac%7B1%7D%7B2%7D%2A+frac%7B1%7D%7B+%28x%5E%7B2%7D+%2B3x+%29%5E%7Bfrac%7B1%7D%7B2%7D%7D+%7D+%2A+%282x%2B3%29%3D+frac%7B2x%2B3%7D%7B2+sqrt%7Bx%5E%7B2%7D+%2B3x%7D+%7D++)
3.
![[ ln( x^{4} + x^{2} ) ]' = frac{( x^{4} + x^{2} )'}{ x^{4} + x^{2} } = frac{4x^3+2x}{ x^{4} + x^{2} } [ ln( x^{4} + x^{2} ) ]' = frac{( x^{4} + x^{2} )'}{ x^{4} + x^{2} } = frac{4x^3+2x}{ x^{4} + x^{2} }](https://tex.z-dn.net/?f=%5B+ln%28++x%5E%7B4%7D+%2B+x%5E%7B2%7D+%29+%5D%27+%3D++frac%7B%28+x%5E%7B4%7D+%2B+x%5E%7B2%7D+%29%27%7D%7B+x%5E%7B4%7D+%2B+x%5E%7B2%7D+%7D+%3D++frac%7B4x%5E3%2B2x%7D%7B+x%5E%7B4%7D+%2B+x%5E%7B2%7D+%7D+)
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