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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 42
Chapter 3, Problem 42

Calculate the derivative of the following functions.
y = cos4 θ + sin4 θ

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Step 1: Recognize that the function y = \(\cos\)^4 \(\theta\) + \(\sin\)^4 \(\theta\) is a sum of two terms, each raised to the fourth power. We will need to use the chain rule to differentiate each term separately.
Step 2: Apply the chain rule to the first term \(\cos\)^4 \(\theta\). The chain rule states that if you have a function u(\(\theta\)) raised to a power n, the derivative is n * u(\(\theta\))^{n-1} * u'(\(\theta\)). Here, u(\(\theta\)) = \(\cos\) \(\theta\) and n = 4.
Step 3: Differentiate \(\cos\) \(\theta\) with respect to \(\theta\) to find u'(\(\theta\)). The derivative of \(\cos\) \(\theta\) is -\(\sin\) \(\theta\).
Step 4: Combine the results from Steps 2 and 3 to find the derivative of \(\cos\)^4 \(\theta\). It will be 4 * \(\cos\)^3 \(\theta\) * (-\(\sin\) \(\theta\)).
Step 5: Repeat Steps 2-4 for the second term \(\sin\)^4 \(\theta\). The derivative of \(\sin\) \(\theta\) is \(\cos\) \(\theta\), so the derivative of \(\sin\)^4 \(\theta\) is 4 * \(\sin\)^3 \(\theta\) * \(\cos\) \(\theta\). Finally, add the derivatives of both terms to get the derivative of the entire function.

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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 value 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 can be calculated using various rules, such as the power rule, product 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 that is the composition of two functions, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions raised to a power, as seen in the given problem.
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Intro to the Chain Rule

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental functions in calculus that relate angles to ratios of sides in right triangles. They are periodic functions and have specific derivatives: the derivative of sin(θ) is cos(θ), and the derivative of cos(θ) is -sin(θ). Understanding these derivatives is essential for calculating the derivative of functions involving trigonometric expressions.
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Related Practice
Textbook Question

Derivative calculations Evaluate the derivative of the following functions at the given point.

f(t) = 1/t+1; a=1

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Textbook Question

Fish length Assume the length L (in centimeters) of a particular species of fish after t years is modeled by the following graph. <IMAGE>

b. What does the derivative tell you about how this species of fish grows?

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Textbook Question

Velocity from position The graph of s=f(t)s = f(t) represents the position of an object moving along a line at time t≥0 t≥0 . <IMAGE>

a. Assume the velocity of the object is 0 when t=0 t=0 . For what other values of t is the velocity of the object zero?

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Textbook Question

Calculate the derivative of the following functions.

y = ⁴√(2x / (4x - 3))

Textbook Question

Velocity from position The graph of s=f(t)s = f(t) represents the position of an object moving along a line at time t≥0t≥0 . <IMAGE>

c. Sketch a graph of the velocity function.

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Textbook Question

Derivatives and tangent lines

a. For the following functions and values of a, find f′(a).

f(x) = 1/3x-1; a= 2