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 3.10.62b
Chapter 3, Problem 3.10.62b

62–65. {Use of Tech} Graphing f and f'
b. Compute and graph f'.
f(x) = (x−1) sin^−1 x on [−1,1]

Verified step by step guidance
1
First, understand the function f(x) = (x - 1) * sin^(-1)(x). This function is defined on the interval [-1, 1] because the inverse sine function, sin^(-1)(x), is only defined for x in [-1, 1].
To find the derivative f'(x), apply the product rule. The product rule states that if you have a function h(x) = u(x) * v(x), then h'(x) = u'(x) * v(x) + u(x) * v'(x). Here, let u(x) = x - 1 and v(x) = sin^(-1)(x).
Calculate the derivatives: u'(x) = 1 and v'(x) = 1 / sqrt(1 - x^2). The derivative of sin^(-1)(x) is 1 / sqrt(1 - x^2).
Apply the product rule: f'(x) = (x - 1) * (1 / sqrt(1 - x^2)) + 1 * sin^(-1)(x). Simplify this expression to get the final form of f'(x).
Use graphing technology to plot both f(x) and f'(x) on the interval [-1, 1]. Observe the behavior of the function and its derivative, noting any critical points or changes in concavity.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
9m
Was this helpful?

Key Concepts

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

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, computing the derivative f' of the function f(x) = (x−1) sin^−1 x will provide insights into the function's behavior, such as its increasing or decreasing nature.
Recommended video:
05:44
Derivatives

Graphing Functions

Graphing a function involves plotting its output values against its input values on a coordinate plane. For the function f(x) = (x−1) sin^−1 x, this means calculating f(x) for various x values within the interval [-1, 1] and representing these points visually. Understanding how to graph both f and its derivative f' helps in analyzing the function's characteristics, such as local maxima and minima.
Recommended video:
5:53
Graph of Sine and Cosine Function

Inverse Sine Function

The inverse sine function, denoted as sin^−1 x or arcsin x, is the function that returns the angle whose sine is x. It is defined for x in the range [-1, 1], producing outputs in the range [-π/2, π/2]. In the given function f(x), the presence of sin^−1 x means that the behavior of f will be influenced by the properties of the inverse sine function, particularly its shape and limits within the specified interval.
Recommended video:
4:03
Inverse Sine
Related Practice
Textbook Question

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

x⁴+y⁴ = 2;(1,−1)

1
views
Textbook Question

109-112 {Use of Tech} Calculating limits The following limits are the derivatives of a composite function g at a point a.

b. Use the Chain Rule to find each limit. Verify your answer by using a calculator.

limh→013((1+h)5+7)10−13(8)10h{\(\displaystyle\]\lim\)_{h\(\to\)0}}\(\frac{\frac{1}{3\left(\left(1+h\right)^5+7\right)^{10}\)}-\(\frac{1}{3\left(8\right)^{10}\)}}{h}

1
views
Textbook Question

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

b. When does the stone reach its highest point?

1
views
Textbook Question

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

b. Use the formula in (a) to find d/dx(e^x(x−1)(x+3))

1
views
Textbook Question

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>


b. What is the rate of change of θ with respect to the bird’s height when it is 60 ft above the ground?

2
views
Textbook Question

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

b. Determine an equation of the line tangent to the curve at the given point.

(x²+y²)²=25/4 xy²; (1, 2)