Problem: Derivative of the inverse function

Problem

We know that if \(y=ln x\) then \(x=e^y\) ( \(x>0)\) , \(ln x\) is the inverse function of \(e^y\) .

Let’s use this to show that the derivative of \(ln x\) is indeed \(1/x\) .

So I am looking for \(dy/dx\) . We know that

We can use

\[ \frac{dy}{dx}= \frac{1}{ \frac{dx}{dy}}\]

1. If \(x=e^y\) , find \(dx/dy\)
2. Find \(1/(dx/dy) \)
3. Rewrite the expression in b) in terms of \(x\) , this is \(dy/dx\) , the derivative of \(log x\) wrt \(x\) .