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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 33
Chapter 3, Problem 33

Calculate the derivative of the following functions.
y = sec(3x+1)

Verified step by step guidance
1
Step 1: Identify the outer function and the inner function. Here, the outer function is \( \sec(u) \) and the inner function is \( u = 3x + 1 \).
Step 2: Recall the derivative of the secant function. The derivative of \( \sec(u) \) with respect to \( u \) is \( \sec(u) \tan(u) \).
Step 3: Apply the chain rule. The chain rule states that the derivative of a composite function \( y = f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).
Step 4: Differentiate the inner function \( u = 3x + 1 \) with respect to \( x \). The derivative \( \frac{du}{dx} \) is 3.
Step 5: Combine the results using the chain rule. The derivative of \( y = \sec(3x + 1) \) is \( \sec(3x + 1) \tan(3x + 1) \cdot 3 \).

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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 changes. It is a fundamental concept in calculus that represents the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as f'(x) or dy/dx, and it can be calculated using various rules, such as the power rule, product rule, quotient rule, and chain rule.
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Derivatives

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 y = f(g(x)), the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is essential when differentiating functions that are nested within each other, such as trigonometric functions with linear transformations.
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Intro to the Chain Rule

Secant Function

The secant function, denoted as sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). It is important in calculus, especially when dealing with derivatives of trigonometric functions. The derivative of sec(x) is sec(x)tan(x), and understanding this relationship is crucial for differentiating functions that involve secant.
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Related Practice
Textbook Question

Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = √(x + 3); P (1,2)

Textbook Question

Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function D(p)=40−2pD(p) = 40-2p , where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

a. According to the model, how many DVDs can be sold in a day at a price of \$10?

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Textbook Question

Equations of tangent lines by definition (2)

b. Determine an equation of the tangent line at P.

f(x) = √x+3; P (1,2)

Textbook Question

Find and simplify the derivative of the following functions.

h(x) = (x − 1)(x3+ x2 + x+1)

Textbook Question

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

For what prices is the demand elastic? Inelastic?

Textbook Question

Based on sales data over the past year, the owner of a DVD store devises the demand function D(p) = 40 - 2p, where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

Find the elasticity function for this demand function.