Originally Posted by
johnsomeone These are both about using the chain rule.
I'll use a function other than the square root to give you an idea.
$\displaystyle \text{A) Suppose } h(x) = [f(x)]^3$
$\displaystyle \text{Then } h'(x) = 3[f(x)]^2f'(x)$
$\displaystyle \text{so } h'(1) = 3[f(1)]^2f'(1) = 3[4]^2(3) = (9)(16) = 144.$
$\displaystyle \text{B) Suppose } \alpha(x) = f(x^3)$
$\displaystyle \text{Then } \alpha'(x) = f'(x^3)(3x^2)$
$\displaystyle \text{so } \alpha'(1) = f'(1^3)(3(1^2)) = 3f'(1) = (3)(3) = 9.$