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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.R.84b
Chapter 3, Problem 3.R.84b

Use the given graphs of f and g to find each derivative. <IMAGE>
b. d/dx (f(x)g(x)) |x=1

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To find the derivative of the product of two functions, f(x) and g(x), at a specific point, we use the product rule. The product rule states that the derivative of a product of two functions is given by: (d/dx)[f(x)g(x)] = f'(x)g(x) + f(x)g'(x).
Evaluate the derivative at x = 1. This means we need to find f'(1), g(1), f(1), and g'(1) from the given graphs.
First, find f'(1) from the graph of f. This is the slope of the tangent line to the curve of f at x = 1.
Next, find g(1) from the graph of g. This is the value of the function g at x = 1.
Similarly, find f(1) from the graph of f and g'(1) from the graph of g. Use these values in the product rule formula to find the derivative at x = 1: f'(1)g(1) + f(1)g'(1).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Product Rule

The Product Rule is a fundamental differentiation rule used to find the derivative of the product of two functions. It states that if you have two functions, f(x) and g(x), the derivative of their product is given by d/dx(f(x)g(x)) = f'(x)g(x) + f(x)g'(x). This rule is essential for solving problems involving the multiplication of functions.
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The Product Rule

Evaluating Derivatives

Evaluating derivatives involves substituting a specific value into the derivative function to find the slope of the tangent line at that point. In this case, you will need to compute the derivatives of f and g at x=1, and then apply the Product Rule to find the derivative of their product at that point. This step is crucial for obtaining the final answer.
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Evaluate Logarithms

Graph Interpretation

Graph interpretation is the ability to analyze and extract information from graphical representations of functions. In this context, understanding the graphs of f and g is vital for determining their values and slopes at x=1. This skill helps in visualizing how the functions behave and aids in accurately applying the Product Rule.
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Related Practice
Textbook Question

9–61. Evaluate and simplify y'.


y = tan^−1 √t²−1

Textbook Question

66–71. Higher-order derivatives Find and simplify y''.


x + sin y = y

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Textbook Question

9–61. Evaluate and simplify y'.


y = 2^x²−x

Textbook Question

{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.

b. Use a graph of L′(t) to describe how the culmen grows over the first 5 weeks of life.

Textbook Question

Use the given graphs of f and g to find each derivative. <IMAGE>

c. d/dx ((f(x) / g(x)) |x=3

Textbook Question

Use differentiation to verify each equation.


d/dx (x⁴ − ln(x⁴ + 1))=4x⁷ / (1 + x⁴).