Skip to main content
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 3.7.25a
Chapter 3, Problem 3.7.25a

Derivatives using tables Let h(x)=f(g(x))h(x)=f(g(x)) and p(x)=g(f(x))p(x)=g(f(x)). Use the table to compute the following derivatives.
<IMAGE>
a. h′(3)h^{\(\prime\)}\(\left\)(3\(\right\))

Verified step by step guidance
1
Step 1: Recognize that the function h(x) = f(g(x)) is a composition of functions, and to find its derivative h'(x), we need to use the chain rule.
Step 2: The chain rule states that if h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x).
Step 3: To find h'(3), substitute x = 3 into the expression for h'(x), giving us h'(3) = f'(g(3)) * g'(3).
Step 4: Use the table to find the values of g(3), f'(g(3)), and g'(3). First, find g(3) from the table.
Step 5: Once you have g(3), use the table to find f'(g(3)) and g'(3), then multiply these values to find h'(3).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function h(x) is composed of two functions f and g, such that h(x) = f(g(x)), then the derivative h'(x) can be found using the formula h'(x) = f'(g(x)) * g'(x). This rule is essential for calculating derivatives of functions that are nested within one another.
Recommended video:
05:02
Intro to the Chain Rule

Derivative Notation

Derivative notation, such as h'(x) or f'(x), represents the rate of change of a function with respect to its variable. It indicates how the function's output changes as its input changes. Understanding this notation is crucial for interpreting and calculating derivatives, especially when dealing with multiple functions and their compositions.
Recommended video:
04:22
Sigma Notation

Function Composition

Function composition occurs when one function is applied to the result of another function. For example, if h(x) = f(g(x)), then g(x) is evaluated first, and its output is used as the input for f. This concept is vital for understanding how to differentiate composite functions and apply the Chain Rule effectively.
Recommended video:
3:48
Evaluate Composite Functions - Special Cases
Related Practice
Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 8x; a = −3

Textbook Question

Use the definition of the derivative to determine d/dx(ax²+bx+c), where a, b, and c are constants.

Textbook Question

60–62. {Use of Tech} Multiple tangent lines Complete the following steps. <IMAGE>

a. Find equations of all lines tangent to the curve at the given value of x.

x+y³−y=1; x=1

Textbook Question

45–50. Tangent lines Carry out the following steps. <IMAGE>

a. Verify that the given point lies on the curve.

x⁴-x²y+y⁴=1; (−1, 1)

Textbook Question

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

g(s) = 4s³ - 8s² +4s / 4s

1
views
Textbook Question

Find an equation of the line tangent to the following curves at the given value of x.

y = csc x; x = π/4