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?

17. \(f\left( x \right) = {e^x}\)

Short answer

The formula for \(f'\left( x \right)\) is \(f'\left( x \right) = {e^x}\).

Step-by-step solution

Sketch the graph of \(f\)

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

Sketch the graph of \(f'\)

Consider two points \(A\) and \(B\) 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 the given function with the above points as shown below:

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 graph of \(f\) is steepest at \(A\) and the graph of \(f\) crosses the \(x\)- axis where \(y = 0\). The slope of the tangent at \(x = A\) is \(0\).

The slope of the function \(f\left( x \right)\) is \(f'\left( B \right)\) 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 \(x = B\). The value of \(f'\left( B \right) = 2.7\).

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

It is observed from the graph of \(f'\left( x \right)\) that the slope at 0 seems to be 1 and the slope at 1 seems to be 2.7. With \(x\) decreases, the slope becomes closer to 0.

We might assume that \(f'\left( x \right) = {e^x}\) because the graph of \(f\) and \(f'\) are so similar.

Thus, the formula for \(f'\left( x \right)\) is \(f'\left( x \right) = {e^x}\).