Inverse Function with Differentiation

[Soaring Crane]

Let f(x) = x^3 + e^x.
Find (f^-1)'(2).
I know how to do everything else except the first step. How do you find the inverse of f(x)? I know the inverse of an exponential function is a logarthmic function, but where do I proceed from here?
Thanks.

 

[benorin]

There's a theorem for that

Let f(x) = x^3 + e^x.
Find (f^-1)'(2).
I know how to do everything else except the first step. How do you find the inverse of f(x)? I know the inverse of an exponential function is a logarthmic function, but where do I proceed from here?
Thanks.

You don't need to find an inverse function to answer this question, you only need to determine the value of its derivative at 2. I am supposing that you know the chain rule:

Let y=g(x) so that g^{-1}(y)=x. Recall that y is a function of x, so in differentiating w.r.t. x we apply the chain rule to get \left( g^{-1} \right)^{\prime}(y)y^{\prime}=1 but y=g(x)
so put y^{\prime}=g^{\prime}(x) and the equation becomes \left( g^{-1} \right)^{\prime}\left( g(x)\right) g^{\prime}(x)=1 or g^{\prime}(x)=\frac{1}{\left( g^{-1} \right)^{\prime}\left( g(x)\right)}.

In your problem, let g(x)=f^{-1}(x).