Chapter 5
The Product, Quotient, and Chain Rules
T
his chapter focuses on some of the major techniques needed to find the derivative: the
product rule, the quotient rule, and the chain rule. By using these rules along with the
power rule and some basic formulas (see Chapter 4), you can find the derivatives of most of
the single-variable functions you encounter in calculus. However, after using the derivative
rules, you often need many algebra steps to simplify the function so that it’s in a nice final
form, especially on problems involving the product rule or quotient rule.
The Problems You’ll Work On
Here you practice using most of the techniques needed to find derivatives (besides the
power rule):
✓ The product rule
✓ The quotient rule
✓ The chain rule
✓ Derivatives involving trigonometric functions
What to Watch Out For
Many of these problems require one calculus step and then many steps of algebraic simplification to get to the final answer. Remember the following tips as you work through the
problems:
✓ Considering simplifying a function before taking the derivative. Simplifying before
taking the derivative is almost always easier than finding the derivative and then
simplifying.
✓ Some problems have functions without specified formulas in the questions; don’t be
thrown off! Simply proceed as you normally would on a similar example.
✓ Many people make the mistake of using the product rule when they should be using the
chain rule. Stop and examine the function before jumping in and taking the derivative.
Make sure you recognize whether the question involves a product or a composition
(in which case you must use the chain rule).
✓ Rewriting the function by adding parentheses or brackets may be helpful, especially
on problems that involve using the chain rule multiple times.
The Product, Quotient, and Chain Rules
T
his chapter focuses on some of the major techniques needed to find the derivative: the
product rule, the quotient rule, and the chain rule. By using these rules along with the
power rule and some basic formulas (see Chapter 4), you can find the derivatives of most of
the single-variable functions you encounter in calculus. However, after using the derivative
rules, you often need many algebra steps to simplify the function so that it’s in a nice final
form, especially on problems involving the product rule or quotient rule.
The Problems You’ll Work On
Here you practice using most of the techniques needed to find derivatives (besides the
power rule):
✓ The product rule
✓ The quotient rule
✓ The chain rule
✓ Derivatives involving trigonometric functions
What to Watch Out For
Many of these problems require one calculus step and then many steps of algebraic simplification to get to the final answer. Remember the following tips as you work through the
problems:
✓ Considering simplifying a function before taking the derivative. Simplifying before
taking the derivative is almost always easier than finding the derivative and then
simplifying.
✓ Some problems have functions without specified formulas in the questions; don’t be
thrown off! Simply proceed as you normally would on a similar example.
✓ Many people make the mistake of using the product rule when they should be using the
chain rule. Stop and examine the function before jumping in and taking the derivative.
Make sure you recognize whether the question involves a product or a composition
(in which case you must use the chain rule).
✓ Rewriting the function by adding parentheses or brackets may be helpful, especially
on problems that involve using the chain rule multiple times.