I was taking a practice clep calculus exam I found online and I do not understand how the correct answer was derived.
$f(x) = x^3 + x$. and $h(x)$ is an inverse function of. find $h'(2)$??? I know that $f(h(x)) = x$ and if $f'(h(x))h'(x) = 1 $, $h'(x) = 1/f'(h(x))$. My understanding is that if $f(2) = 10$ then $h(10) = 2$ because they are inverses. Could somebody help me with my misunderstanding? I think the correct answer was $1/4$.
Sure your understanding is correct but you’re evaluating at the wrong point.
You’re interested in $h’(2)$, so try to evaluate the right hand side $1/f’(h(x))$ at $2$. You need $h(2)$. What value, when plugged into $f$, gives $2$?
More rigorously you should prove that $f$ is invertible, depending on how strict your exam is. But this is easy, since $f$ is strictly increasing and continuous.