Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.6.72

Chapter 3, Problem 3.6.72

Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1

Verified step by step guidance

First, understand that you need to find the derivative of the composite function (f ∘ g)(x), which means you need to apply the chain rule. The chain rule states that (f ∘ g)'(x) = f'(g(x)) * g'(x).

Calculate g(x) at x = -1. Substitute x = -1 into g(x) = (1 / x²) - 1 to find g(-1).

Next, find g'(x). Differentiate g(x) = (1 / x²) - 1 with respect to x. Use the power rule and the derivative of a constant.

Now, find f'(u). First, differentiate f(u) = ((u - 1) / (u + 1))² with respect to u. Use the chain rule and the quotient rule to differentiate this expression.

Finally, substitute g(-1) into f'(u) to find f'(g(-1)), and multiply this by g'(-1) to find (f ∘ g)'(-1).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

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 technique in calculus used to differentiate composite functions. If you have a function h(x) = f(g(x)), the derivative h'(x) is found by multiplying the derivative of the outer function f at g(x) by the derivative of the inner function g at x. This rule is essential for finding derivatives of nested functions like f ∘ g.

Composite Functions

Composite functions involve applying one function to the results of another, denoted as (f ∘ g)(x) = f(g(x)). Understanding how to work with composite functions is crucial for applying the chain rule effectively, as it requires recognizing the inner and outer functions and their respective derivatives.

Derivative of Rational Functions

Rational functions are quotients of polynomials, and their derivatives can be found using the quotient rule. For a function f(u) = (u - 1)/(u + 1), the derivative involves differentiating the numerator and denominator separately and applying the quotient rule: (v'u - uv')/v², where u and v are the numerator and denominator, respectively.

Recommended video:

6:04

Intro to Rational Functions

Related Practice

Textbook Question

The eight curve Find the slopes of the curve y⁴ = y² – x² at the two points shown here.

202

views

Textbook Question

Differential Estimates of Change

In Exercises 35–40, write a differential formula that estimates the given change in volume or surface area.

The change in the lateral surface area S = 2πrh of a right circular cylinder when the height changes from h₀ to h₀ + dh and the radius does not change

138

views

Textbook Question

In Exercises 41–58, find dy/dt.

y = sin²(πt − 2)

272

views

Textbook Question

Find the derivatives of the functions in Exercises 1–42.

s = (sec t + tan t)⁵

293

views