I took notes in class, but am totally stuck now. I need to find the inverse function of f(x)=4x+10x^9 at c= -14
no, you don't need the inverse of $\displaystyle f(x) = 4x + 10x^9$.
if $\displaystyle g(x)$ is the inverse of $\displaystyle f(x)$, then $\displaystyle f[g(x)] = x$
$\displaystyle \frac{d}{dx}(f[g(x)] = x)$
$\displaystyle f'[g(x)] \cdot g'(x) = 1$
$\displaystyle g'(x) = \frac{1}{f'[g(x)]}$
note that $\displaystyle f(-1) = -14$
since $\displaystyle g(x)$ is the inverse, $\displaystyle g(-14) = -1$
so ...
$\displaystyle g'(-14) = \frac{1}{f'[g(-14)]}$
finish up and find $\displaystyle g'(-14)$.
Alternatively, remember that for an inverse, the domain and ranges swap. I.e, the $\displaystyle x$ and $\displaystyle y$ values swap.
So if $\displaystyle f(x) = y = 4x + 10x^9$ then $\displaystyle f^{-1}(x) = x = 4y + 10y^9$.
You can differentiate the inverse using implicit differentiation.
$\displaystyle x = 4y + 10y^9$
$\displaystyle \frac{d}{dx}(x) = \frac{d}{dx}(4y + 10y^9)$
$\displaystyle 1 = \frac{dy}{dx}\,\frac{d}{dy}(4y + 10y^9)$
$\displaystyle 1 = \frac{dy}{dx}(4 + 90y^8)$
$\displaystyle \frac{dy}{dx} = \frac{1}{4 + 90y^8}$.
It'd be really tough to write the derivative in terms of $\displaystyle x$ though, so you don't bother.
If you're trying to evaluate this derivative at $\displaystyle x = -14$ then find what $\displaystyle y$ equals there, then substitute the $\displaystyle y$ value into the derivative.