найдите производную функции 1)y=(x^2-5x)(1-2 корень x)
2)y=2/корень x
3)y=-2/cos x
4)y=sin x/(x^2+3)
5)y=(3x+1)ctg x
Ответы
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1) ![y'(x)=(x^{2}-5x )'*(1-2 sqrt{x} )+(x^{2}-5x )*(1-2 sqrt{x} )'= y'(x)=(x^{2}-5x )'*(1-2 sqrt{x} )+(x^{2}-5x )*(1-2 sqrt{x} )'=](https://tex.z-dn.net/?f=y%27%28x%29%3D%28x%5E%7B2%7D-5x+%29%27%2A%281-2+sqrt%7Bx%7D+%29%2B%28x%5E%7B2%7D-5x+%29%2A%281-2+sqrt%7Bx%7D+%29%27%3D)
![= (2x-5)*(1-2 sqrt{x} )+(x^{2}-5x )*(-frac{1}{sqrt{x}} ) = (2x-5)*(1-2 sqrt{x} )+(x^{2}-5x )*(-frac{1}{sqrt{x}} )](https://tex.z-dn.net/?f=%3D+%282x-5%29%2A%281-2+sqrt%7Bx%7D+%29%2B%28x%5E%7B2%7D-5x+%29%2A%28-frac%7B1%7D%7Bsqrt%7Bx%7D%7D+%29)
2)![y=2 x^{- frac{1}{2}} y=2 x^{- frac{1}{2}}](https://tex.z-dn.net/?f=y%3D2++x%5E%7B-+frac%7B1%7D%7B2%7D%7D+)
![y'(x)= -frac{2}{2} x^{- frac{3}{2} }= -x^{- frac{3}{2} } y'(x)= -frac{2}{2} x^{- frac{3}{2} }= -x^{- frac{3}{2} }](https://tex.z-dn.net/?f=y%27%28x%29%3D+-frac%7B2%7D%7B2%7D+x%5E%7B-+frac%7B3%7D%7B2%7D+%7D%3D+-x%5E%7B-+frac%7B3%7D%7B2%7D+%7D+)
3)![y=-2Cos^{-1}x y=-2Cos^{-1}x](https://tex.z-dn.net/?f=y%3D-2Cos%5E%7B-1%7Dx+)
![y'(x)=2Sinx*Cos^{-2}x= frac{2Sinx}{Cos^{2}x} y'(x)=2Sinx*Cos^{-2}x= frac{2Sinx}{Cos^{2}x}](https://tex.z-dn.net/?f=y%27%28x%29%3D2Sinx%2ACos%5E%7B-2%7Dx%3D+frac%7B2Sinx%7D%7BCos%5E%7B2%7Dx%7D+)
4)![y'(x)= frac{(Sinx)'(x^{2}+3)-Sinx*( x^{2} +3)'}{( x^{2} +3)^{2} } = y'(x)= frac{(Sinx)'(x^{2}+3)-Sinx*( x^{2} +3)'}{( x^{2} +3)^{2} } =](https://tex.z-dn.net/?f=y%27%28x%29%3D+frac%7B%28Sinx%29%27%28x%5E%7B2%7D%2B3%29-Sinx%2A%28+x%5E%7B2%7D+%2B3%29%27%7D%7B%28+x%5E%7B2%7D+%2B3%29%5E%7B2%7D+%7D+%3D)
![frac{Cosx*( x^{2} +3)-2x*Sinx}{( x^{2} +3)^{2} } frac{Cosx*( x^{2} +3)-2x*Sinx}{( x^{2} +3)^{2} }](https://tex.z-dn.net/?f=+frac%7BCosx%2A%28+x%5E%7B2%7D+%2B3%29-2x%2ASinx%7D%7B%28+x%5E%7B2%7D+%2B3%29%5E%7B2%7D+%7D+)
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