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 96c
Chapter 3, Problem 96c

Composition containing sin x Suppose f is differentiable on [−2,2] with f′(0)=3 and f′(1)=5. Let g(x)=f(sin x). Evaluate the following expressions.
c. g'(π)

Verified step by step guidance
1
Step 1: Recognize that g(x) is a composition of functions, where g(x) = f(sin x). To find g'(x), we need to use the chain rule for differentiation.
Step 2: The chain rule states that if you have a composition of functions h(x) = f(u(x)), then the derivative h'(x) = f'(u(x)) * u'(x).
Step 3: In this problem, u(x) = sin x, so u'(x) = cos x. Therefore, g'(x) = f'(sin x) * cos x.
Step 4: We need to evaluate g'(π). First, find sin(π), which is 0. So, g'(π) = f'(sin(π)) * cos(π).
Step 5: Substitute the known values: f'(0) = 3 and cos(π) = -1. Therefore, g'(π) = f'(0) * (-1).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
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 principle in calculus used to differentiate composite functions. It states that if you have a function g(x) = f(h(x)), the derivative g'(x) can be found by multiplying the derivative of the outer function f with the derivative of the inner function h. This rule is essential for evaluating derivatives of functions that are composed of other functions.
Recommended video:
05:02
Intro to the Chain Rule

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In this context, sin(x) is used as the inner function in the composition g(x) = f(sin x). Understanding the behavior and derivatives of these functions is crucial for evaluating expressions involving them, especially at specific points like π.
Recommended video:
6:04
Introduction to Trigonometric Functions

Differentiability

A function is said to be differentiable at a point if it has a defined derivative there, which implies that the function is smooth and continuous at that point. In this problem, knowing that f is differentiable on the interval [−2,2] allows us to apply the Chain Rule confidently, as we can use the given derivatives f′(0) and f′(1) to find g'(π) through the composition of functions.
Recommended video:
05:53
Finding Differentials
Related Practice
Textbook Question

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>

Explain why the average rate of growth from 1950 to 1960 is a good approximation to the (instantaneous) rate of growth in 1955.

Textbook Question

Product Rule for three functions Assume f, g, and h are differentiable at x.

a. Use the Product Rule (twice) to find a formula for d/dx (f(x)g(x)h(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.


a. Find L′(1) and interpret the meaning of this value.

Textbook Question

Given that p(x) = (5e^x+10x⁵+20x³+100x²+5x+20) ⋅ (10x⁵+40x³+20x²+4x+10), find p′(0) without computing p′(x).

Textbook Question

Composition containing sin x Suppose f is differentiable for all real numbers with f(0)=−3,f(1)=3,f′(0)=3, and f′(1)=5. Let g(x)=sin(πf(x)). Evaluate the following expressions.

b. g'(1)

Textbook Question

Tangent lines Assume f is a differentiable function whose graph passes through the point (1, 4). Suppose g(x)=f(x²) and the line tangent to the graph of f at (1, 4) is y=3x+1. Find each of the following.

a. g(1)