of the derivative."/>
Estimate the value of f′(3) using the graph.
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Estimate the value of f′(–1) using the graph.
293.
Estimate the value of f′(–3) using the graph.
294.
Based on the graph of
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Based on the graph, arrange the following from smallest to largest: f′(–3), f′(–2), and f′(1).
296.
Based on the graph, arrange the following from smallest to largest: f′(1), f′(2), f′(5), and 0.1.
Using the Power Rule to Find Derivatives
297–309 Use the power rule to find the derivative of the given function.
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Finding All Points on a Graph Where Tangent Lines Have a Given Value
310–311 Find all points on the given function where the slope of the tangent line equals the indicated value.
310. Find all x values where the function
311. Find all x values where the function
Chapter 5
The Product, Quotient, and Chain Rules
This 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.
Using the Product Rule to Find Derivatives
312–331 Use the product rule to find the derivative of the given function.
312.
