Question

Make a careful sketch of the graph of \(f\) and below it sketch the graph \(f'\) in the same manner as in Exercise 4-11. Can you guess a formula for \(f'\left( x \right)\) from its graph?

18. \(f\left( x \right) = \ln x\)

Short answer

The formula for \(f'\left( x \right)\) from its graph is \(f'\left( x \right) = \frac{1}{{{x^2}}}\) or \(f'\left( x \right) = \frac{1}{x}\).

Step-by-step solution

Sketch the graph of \(f\)

Consider the point \(A\) to plot on the graph of the given function \(f\left( x \right)\). We can estimate the graph of derivatives from these points.

Sketch the graph of \(f\left( x \right) = {e^x}\) as shown below:

Sketch the graph of \(f'\)

To determine the value of the derivative at any value of \(x\), we can draw the tangent at the point \(\left( {x,f\left( x \right)} \right)\) and estimate the slope.

The slope of the function \(f\left( x \right)\) is \(f'\left( A \right)\) can be determined as the ratio of the rise and run that appears to be equal. The function \(f\left( x \right)\) is increasing at \(x = A\). The value of \(f'\left( A \right) = 3\).

Use the above points to sketch the graph of \(f'\left( x \right)\) as shown below:

Guess the formula for \(f'\left( x \right)\) from its graph

With the increases of \(x\) towards 1, the derivative \(f'\left( x \right)\) decreases a very large number to 1. As \(x\) increases, \(f'\left( x \right)\) approaches 0. We might assume that \(f'\left( x \right) = \frac{1}{{{x^2}}}\) or \(f'\left( x \right) = \frac{1}{x}\).
Thus, the formula for \(f'\left( x \right)\) from its graph is \(f'\left( x \right) = \frac{1}{{{x^2}}}\) or \(f'\left( x \right) = \frac{1}{x}\).