The given function is f(x)=ex.

From the graph of f(x), it is clear that the slope is always positive. Thus, the derivative of the graph is always positive.

Draw the tangent to the graph at x=0 and the slope of the graph is computed as follows,

m=0−1−1−0=−1−1=1

Recall the fact that, the slope of the function at the point x=a is equal to the derivative of the function at that point.

Therefore, f′(x)=1 at x=0.

Thus, the graph of the function f′(x) passes through the point (0,1).

From the point A to left, the slope is increasing slowly. Thus the graph of the function f′(x) in this section will rise slowly.

From the point A to right, the slope increases rapidly. Thus the graph of the function f′(x) in this section will rise highly.

Graph:

Use the online graphing calculator to draw the graph of f(x)=ex and use the above information to trace the graph of f′(x) as shown below in Figure 1.

Observation:

From the Figure 1, it is observed that the graph of the derivative function seems to be a natural exponential function.

Thus, the derivative formula is, f′(x)=ex.