Introduction

Recall that the function \( f(x) = e^x \) has the inverse function \( f^{-1}(x) = \ln(x)\).

Let's investigate the slope of the tangent to many points along the curve of \(f(x)=\ln(x)\) by using the following Maple investigation.

In this investigation, you will see the graph of \(f(x)=\ln(x)\) and a tangent drawn at one point on the left side of the graph.

Maple has calculated the slope of this tangent and then plotted the \(x\)-coordinate of the point of tangency with the value of the slope of the tangent at that point.

This will give us the numerical values of the derivative of \(f(x)=\ln(x)\).

Try it now. If we were to connect these values, what function do you see?