Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post some work that I've tried, but I'm not sure where to even start. My professor is bad at explaining things so I'm turning to the site for supplemental instruction.
"For each of the given functions $f(x)$, find the derivative $\left. \frac{d}{dx}f^{-1}(x) \right|_{x=c}$ at the given point $c$, first finding $a=f^{-1}(c)$. $$ f(x)= 3 x + 9 x^{13}; \ \ c = -12, \ \ \ a=? \ \ \ (f^{-1})'(c)=? $$
$f(x)= x^2 - 13 x + 58$ on the interval $[6.5,\infty);$ $c = 18$, $a=?$ $(f^{-1})'(c) = ?$"
I've been able to find the inverse of other functions not involving higher powers like the $9x^{13}$, and then finding the derivative of that inverse, but this particular kind of question stumps me.
For the first one, note that if $a = f^{-1}(-12)$ then $$ 3a + 9a^{13} = f(a) = -12. $$ You don't have to find the inverse of the entire function here, only the preimage of that one point. If you take $a=-1$ then you will have a solution to that equation, therefore $f^{-1}(-12) = -1$.
Also notice $(f^{-1})'(c) = (f^{-1})'(f(a)) = 1/f'(a)$ by your equation, so you don't need to find the inverse for the second part of the question either!