Differentiation - exponential functions

Investigate exponential functions of the form y = Ma^kx with this Exploremath activity.

Consider exponential functions of the form y = a^x for different values of a greater than 0.

Use the graphics calculator to draw the graph of the exponential function.

Also draw a secant gradient function with a small x-increment (say 0.001). The secant gradient function will be a good approximation to the derivative.

For large values of a, the graph of the derivative is above the graph of the original function.

For small values of a, the graph of the derivative is below the graph of the original function.

The number e is defined as the value of a for which the derivative is identical to the exponential function.

At any point on the graph of  y = e^x , the value of the derivative is equal to the value of y.

To differentiate any exponential function with base e, use the chain rule:

To differentiate an exponential function with a base other than e, use the following result (together with the chain rule if necessary):

This result is justified as follows:

NB. log_e is written ln (meaning natural logarithm).