Question

Derivative Function.. Could you solve this, please?

Answer 1

Hello, please consider the following.

f is differentiable so g is differentiable and we we can compute the derivative of g.

`\text{derivative of } u^2\text{ is } 2uu'\\\\\text{derivative of } f(f(x^3)+x)\text{ is } (3x^2f'(x^3)+1)f'(f(x^3)+x)\\\\\text{So, }\\\\g'(x)=2f(f(x^3)+x)(3x^2f'(x^3)+1)f'(f(x^3)+x)\\\\\text{ We replace x par 1}\\\\g'(1)=2f(f(1)+1)(3f'(1)+1)f'(f(1)+1)\\\\g'(1)=2f(2)(3f'(1)+1)f'(2)\\\\\text{ We can put f'(2) in one part of the equation}\\\\f'(2)=(2f(2)(3f'(1)+1))/(g'(1))=(2*5*(3*(-3)+1))/(2)\\\\=5*(-9+1)\\\\=5*(-8)\\\\=\boxed{-40}`

Thank you.