Intuitive way to think about the derivative of an inverse function

Question

I know that one can bring out the formula for the derivative of an inverse function using the chain rule, but is there an intuitive way to think about it? I can't seem to grasp it too well.

Answer 1

Inverse of a function is $$f^{-1}[f(x)]=x,f[f^{-1}(x)]=x$$ Let's denote the inverse function of $f$ by $g$ then we have $$g[f(x)]=x$$

Now take derivative on both sides of the above equation and we get,

$$g^{\prime}[f(x)]\cdot f^{\prime}(x)=1$$ Now lets say a variable $y=f(x)$ Replacing this we get $$g^{\prime}(y)\cdot f^{\prime}(x)=1$$ So, $$g^{\prime}(y)=\frac{1}{f^{\prime}(x)}$$