Section 5.4 Exponential Functions: Differentiation and Integration

Since we know how to deal with the natural logarithm function for integrating and differentiating, and we have refreshed ourselves on the issue of inverse functions, it stands to reason that we can discuss the derivative and antiderivative of the natural exponential function

Section 5.4 Exponential Functions: Differentiation and Integration Remember the following relationship: if and only if  Since we know how to take the derivative of a natural logarithm function, we should transform the equations above as follows:

Section 5.4 Exponential Functions: Differentiation and Integration So, what we have with the natural exponential function is a function that is its own derivative. The chain rule leads us to a relatively logical extension that looks like this, with two examples below:

Section 5.4 Exponential Functions: Differentiation and Integration How can we deal with integrals involving functions with a base of e? We need to keep the chain rule in mind at all times with these problems. We will be looking to divide the derivative of the power of the exponential function in each case. Look at the following example:

Section 5.4 Exponential Functions: Differentiation and Integration What if the exponent is more complicated and has a function as its derivative instead of simply a constant? In that case we need to have a product of functions involved in the integral. Look at the two examples below, one of which we can integrate. Decide which one is doable at this point.

Section 5.4 Exponential Functions: Differentiation and Integration Let’s try a couple of examples of definite integrals. Remember, look for substitutions so that you can see the function in the form of

Section 5.4 Exponential Functions: Differentiation and Integration A couple of more examples…