Question

First derivative of this function ?​

Answer 1

Use the chain rule.

$f'(x) = \left[\left(x^2 + \cos^2(x)\right)^3\right]' \\\\ ~~~~ = 3 \left(x^2 + \cos^2(x)\right)^2 \left[x^2 + \cos^2(x)\right]' \\\\ ~~~~ = 3 \left(x^2 + \cos^2(x)\right)^2 \left(\left[x^2\right]' + \left[\cos^2(x)\right]'\right) \\\\ ~~~~ = 3 \left(x^2 + \cos^2(x)\right)^2 \left(2x + 2 \cos(x) \left[\cos(x)\right]'\right) \\\\ ~~~~ = 3 \left(x^2 + \cos^2(x)\right)^2 \left(2x - 2 \cos(x) \sin(x)\right) \\\\ ~~~~ = \boxed{3 \left(x^2 + \cos^2(x)\right)^2 \left(2x - \sin(2x)\right)}$