3.Найдите производные сложных функций.
а) f(х)= (х-1)²(х+1); б) q(х) = sinx/3 - tg²x.
Помогите пожалуйста
Ответы
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a) ![f(x)= (x-1)^2(x+1) f(x)= (x-1)^2(x+1)](https://tex.z-dn.net/?f=+f%28x%29%3D+%28x-1%29%5E2%28x%2B1%29)
![f'(x)= ((x-1)^2)'*(x+1)* (x-1)^2(x+1)'= f'(x)= ((x-1)^2)'*(x+1)* (x-1)^2(x+1)'=](https://tex.z-dn.net/?f=+f%27%28x%29%3D+%28%28x-1%29%5E2%29%27%2A%28x%2B1%29%2A+%28x-1%29%5E2%28x%2B1%29%27%3D)
![=2(x-1)*(x-1)'*(x+1)+(x-1)^2*1= =2(x-1)*(x-1)'*(x+1)+(x-1)^2*1=](https://tex.z-dn.net/?f=%3D2%28x-1%29%2A%28x-1%29%27%2A%28x%2B1%29%2B%28x-1%29%5E2%2A1%3D)
![=2(x-1)(x+1)+(x-1)^2=2(x^2-1)+x^2-2x+1= =2(x-1)(x+1)+(x-1)^2=2(x^2-1)+x^2-2x+1=](https://tex.z-dn.net/?f=%3D2%28x-1%29%28x%2B1%29%2B%28x-1%29%5E2%3D2%28x%5E2-1%29%2Bx%5E2-2x%2B1%3D)
![=2x^2-2+x^2-2x+1=3x^2-2x-1 =2x^2-2+x^2-2x+1=3x^2-2x-1](https://tex.z-dn.net/?f=%3D2x%5E2-2%2Bx%5E2-2x%2B1%3D3x%5E2-2x-1)
б)![q(x) = sin frac{x}{3} - tg^2x. q(x) = sin frac{x}{3} - tg^2x.](https://tex.z-dn.net/?f=q%28x%29+%3D+sin+frac%7Bx%7D%7B3%7D++-+tg%5E2x.)
![q'(x) =( sin frac{x}{3})'- (tg^2x)'=cos frac{x}{3} *( frac{x}{3} )'-2tgx*(tgx)'= q'(x) =( sin frac{x}{3})'- (tg^2x)'=cos frac{x}{3} *( frac{x}{3} )'-2tgx*(tgx)'=](https://tex.z-dn.net/?f=q%27%28x%29+%3D%28+sin+frac%7Bx%7D%7B3%7D%29%27-+%28tg%5E2x%29%27%3Dcos+frac%7Bx%7D%7B3%7D+%2A%28+frac%7Bx%7D%7B3%7D+%29%27-2tgx%2A%28tgx%29%27%3D)
![=cos frac{x}{3}* frac{1}{3} -2tgx*frac{1}{cos^2x} = frac{cos frac{x}{3}}{3}- frac{2tgx}{cos^2x} =cos frac{x}{3}* frac{1}{3} -2tgx*frac{1}{cos^2x} = frac{cos frac{x}{3}}{3}- frac{2tgx}{cos^2x}](https://tex.z-dn.net/?f=%3Dcos+frac%7Bx%7D%7B3%7D%2A+frac%7B1%7D%7B3%7D+-2tgx%2Afrac%7B1%7D%7Bcos%5E2x%7D+%3D+frac%7Bcos+frac%7Bx%7D%7B3%7D%7D%7B3%7D-+frac%7B2tgx%7D%7Bcos%5E2x%7D+)
б)
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