Suppose we are given an exponential function f(x) = ax. We can use the definition of the derivative to find the derivative of that function.

Notice that the limit term is equivalent to the derivative of the function evaluated at zero.

Therefore, the derivative of the exponential function is proportional to the function itself.

We discussed the natural exponential function ex in an earlier lesson; a graph of the function is shown here with a tangent line touching the curve. Use the slider to control the x value of the tangent point.

What is the value of f′ (0)? Click the "Submit" button after entering your answer.

We found that the derivative f′ (0) of the natural exponential function ex is equal to one. Now, let's substitute this value into the formula we derived for the derivative of an exponential function.

This gives us a rule for determining the derivative of natural exponential functions. The derivative of a natural exponential function is just the function itself.

We can confirm this result by experimenting with the graph of the natural exponential function and the tangent slope. Use the slider on the graph to vary the value of x. Notice that the slope of the tangent and the value of the function at the tangent point are always equal.

Just as we found possible with other differentiation rules, we can combine the differentiation rule for the natural exponential function and the chain rule. Suppose we have a function given by eu where u is a function of x.

Applying the chain rule followed by the natural exponential rule yields the natural exponential function rule form of the chain rule.

We can use this new form of the chain rule to determine the derivative of the more complex function shown here.

As always, the first step in using the chain rule is deciding what to choose for the variable u. Click the "Submit" button after selecting your answer.

To determine the derivative using the chain rule, we set u = sin x. The next step is to determine the derivative of u with respect to x.

What is the derivative du/dx? Click the "Submit" button after entering your answer.

Now that we have an expression for the derivative du/dx, we can solve for the derivative of the original function.

The natural exponential function can be used to model real-world problems, such as the spread of a contagious disease. The function shown here represents the number of people infected by a disease, with respect to time t in days.

What is the rate of infection after 4 days? Click the "Submit" button after entering your answer.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education