In section 11.5, you learned how to take the derivative of a product. For a product of two function of the variable x, called u and v, the derivative of the product is
In this form, the variable on each function has been left off to make the rule easier to remember.
In this section, we’ll learn how to reverse the product rule. This technique is called integration by parts. It helps you to find the antiderivative of functions that are written as a product.
In Chapter 13, we reversed the derivative process for basic functions like power and exponential functions. By taking the antiderivative of a power function, we were able to find the original function we had taken the derivative of. In this question, we continue to find antiderivatives of function. For example, suppose we want to take the derivative
Since the express is a composition of two functions, we must use the chain rule to take this derivative. Start by identifying the inside and outside functions of the composition,
The derivatives of these functions are
Using the chain rule,
This derivative can also be written as the antiderivative,
If we have already carried out the derivative, writing out the antiderivative is just the reverse process. However, it is rarely the case that we have the derivative available to help us evaluate the antiderivative. For integrands involving compositions, Substitution Method may help us to find the antiderivative.
In this chapter, we have examined how to estimate the change in a function from its derivative. Using a table of values of the derivative or the derivative of a function itself, we have estimated how the original function changes. This connection involves area and antiderivatives. In this section, we make this more concrete by introducing the Fundamental Theorem of Calculus. This useful theorem allows us to calculate the value of a definite integral using the antiderivative of the integrand.
In this section we continue to compute areas using rectangles. Doing this allows us to undo the process of taking the derivative of a function. From the derivative of a function, we can estimate changes in the function and make the estimates arbitrarily close.
By using more and more rectangles over an interval, the area of the rectangles approach the area between the derivative function and the x axis over the interval. This area is the exact change in the original function and is represented by a definite integral.
How a quantity changes is a very important concept in business and economics. In earlier chapters, we examined several functions like the revenue, cost and profit functions. From these functions, we are able to calculate the rate at which each function changes. The mathematical concept of a derivative helped us to understand how quantitative decisions may be made about these functions.
In this section, we reverse this process. Starting from the rate of change of some function, we’ll calculate information about the function itself. For instance, by knowing the rate of change of revenue, we can calculate how revenue changes when production is changed. In this context, area becomes very important. We’ll examine how area relates to this process and extend the concept to calculate the area under a function.
