Question

How do you calculate the derivative of the function `f(x)=cos(x^3+x^2+1)`?

Answer 1

Use the chain rule.

`d/dx(f(g(x))=f'(g(x))g'(x)`

In this problem, the outermost function (`f`) is the cosine function.

The derivative of the cosine function is the opposite of the sign function.

So we take `f'` (the opposite of the sine) evaluated at the inside function (`g(x)` which is `x^3+x^2+1`) TIMES the derivative of the inside function.

The derivative is `-sin(x^3+x^2+1)` times `(3x^2+2x)` (the derivative of the inside function).

`f'(x)=(-sin(x^3+x^2+1))*(3x^2+2x)=-(3x^2+2x)sin(x^3+x^2+1)`.