1. ## Differentiating the inverse

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

2. Originally Posted by bnation
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 $f(x) = 4x + 10x^9$.

if $g(x)$ is the inverse of $f(x)$, then $f[g(x)] = x$

$\frac{d}{dx}(f[g(x)] = x)$

$f'[g(x)] \cdot g'(x) = 1$

$g'(x) = \frac{1}{f'[g(x)]}$

note that $f(-1) = -14$

since $g(x)$ is the inverse, $g(-14) = -1$

so ...

$g'(-14) = \frac{1}{f'[g(-14)]}$

finish up and find $g'(-14)$.

3. Originally Posted by bnation
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
Alternatively, remember that for an inverse, the domain and ranges swap. I.e, the $x$ and $y$ values swap.

So if $f(x) = y = 4x + 10x^9$ then $f^{-1}(x) = x = 4y + 10y^9$.

You can differentiate the inverse using implicit differentiation.

$x = 4y + 10y^9$

$\frac{d}{dx}(x) = \frac{d}{dx}(4y + 10y^9)$

$1 = \frac{dy}{dx}\,\frac{d}{dy}(4y + 10y^9)$

$1 = \frac{dy}{dx}(4 + 90y^8)$

$\frac{dy}{dx} = \frac{1}{4 + 90y^8}$.

It'd be really tough to write the derivative in terms of $x$ though, so you don't bother.

If you're trying to evaluate this derivative at $x = -14$ then find what $y$ equals there, then substitute the $y$ value into the derivative.