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