1. ## Derivative of Inverse

If f(X) = x+2e^x, find the value of g'(1+2e), where g(x) = inverse of f(x) for all x

I have no idea where to start. I tried to find the inverse of f(x) but that only led to x=y+2e^y and I'm stuck there.

2. Originally Posted by vnsin

If f(X) = x+2e^x, find the value of g'(1+2e), where g(x) = inverse of f(x) for all x

I have no idea where to start. I tried to find the inverse of f(x) but that only led to x=y+2e^y and I'm stuck there.

Let $y = f^{-1}(x) \Rightarrow f(y) = x$. Therefore:
$y + 2 e^y = x$
$\Rightarrow \frac{dy}{dx} + 2e^y \frac{dy}{dx} = 1 \Rightarrow \frac{dy}{dx} (1 + 2 e^y) = 1 \Rightarrow \frac{dy}{dx} = \frac{1}{1 + 2 e^y}$.
Now note that $f(x) = 1 + 2 e \Rightarrow x = 1$. Therefore $y = f^{-1}(1 + 2e) = 1$.
Substitute $y = 1$ into $\frac{dy}{dx} = \frac{1}{1 + 2 e^y}$:
$\frac{dy}{dx} = \frac{1}{1 + 2e}$.