Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.10.40

Chapter 3, Problem 3.10.40

13–40. Evaluate the derivative of the following functions.
f(x) = 1/tan^−1(x²+4)

Verified step by step guidance

Step 1: Recognize that the function f(x) = \(\frac{1}{\tan^{-1}\)(x^2 + 4)} is a composition of functions, where the outer function is g(u) = \(\frac{1}{u}\) and the inner function is u(x) = \(\tan\)^{-1}(x^2 + 4).

Step 2: Apply the chain rule for differentiation, which states that the derivative of a composite function g(u(x)) is g'(u(x)) * u'(x).

Step 3: Differentiate the outer function g(u) = \(\frac{1}{u}\) with respect to u to get g'(u) = -\(\frac{1}{u^2}\).

Step 4: Differentiate the inner function u(x) = \(\tan\)^{-1}(x^2 + 4) with respect to x. Use the derivative formula for \(\tan\)^{-1}(v), which is \(\frac{1}{1+v^2}\), and apply it to v = x^2 + 4. Also, apply the chain rule to differentiate x^2 + 4.

Step 5: Combine the results from Steps 3 and 4 using the chain rule: f'(x) = g'(u(x)) * u'(x). Substitute back u(x) = \(\tan\)^{-1}(x^2 + 4) and simplify the expression.

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.

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the function's graph at any given point. The derivative is often denoted as f'(x) or df/dx and can be calculated using various rules such as the power rule, product rule, and chain rule.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function g(x) that is composed with another function f(u), where u = g(x), then the derivative of the composite function f(g(x)) is f'(g(x)) * g'(x). This rule is essential when differentiating functions that involve nested expressions, such as the function in the given question.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan^−1(x), are the functions that reverse the action of the standard trigonometric functions. For example, tan^−1(x) gives the angle whose tangent is x. When differentiating functions involving inverse trigonometric functions, specific derivative formulas apply, such as the derivative of tan^−1(x) being 1/(1+x²), which is crucial for evaluating the derivative of the given function.

Recommended video:

06:35

Derivatives of Other Inverse Trigonometric Functions

Related Practice

Textbook Question

A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?

303

views

Textbook Question

15–48. Derivatives Find the derivative of the following functions.

P = 40/1+2^-t

232

views

Textbook Question

{Use of Tech} Difference quotients Suppose f is differentiable for all x and consider the function D(x) = f(x+0.01)-f(x) / 0.01 For the following functions, graph D on the given interval, and explain why the graph appears as it does. What is the relationship between the functions f and D?

f(x) = sin x on [−π,π]

268

views

Textbook Question

Find the slope of the curve x²+y³=2 at each point where y=1 (see figure). <IMAGE>

262

views

Textbook Question

Find the derivative of the following functions.

y = x² (1 - In x²)

215

views