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 3.5.64

Chapter 3, Problem 3.5.64

Find y'' for the following functions.
y = cos θ sin θ

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First, recognize that the function y = cos(θ) sin(θ) is a product of two functions. To find the derivative, we will use the product rule, which states that if y = u*v, then y' = u'v + uv'.

Identify u = cos(θ) and v = sin(θ). Compute the derivatives u' and v'. The derivative of cos(θ) is -sin(θ), and the derivative of sin(θ) is cos(θ).

Apply the product rule: y' = (-sin(θ)) * sin(θ) + cos(θ) * cos(θ). Simplify this expression to get y'.

Now, to find y'', differentiate y' again. Use the sum rule and the chain rule as needed. Differentiate each term separately: for the first term, differentiate -sin(θ) * sin(θ), and for the second term, differentiate cos(θ) * cos(θ).

Combine the derivatives from the previous step to obtain y''. Simplify the expression to get the second derivative of y with respect to θ.

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Key Concepts

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

Second Derivative

The second derivative of a function measures the rate of change of the first derivative, providing information about the curvature of the function's graph. It is denoted as y'' and is essential for analyzing the concavity and inflection points of the function. In this context, finding y'' involves differentiating the function twice with respect to the variable.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, especially when dealing with periodic phenomena. The function y = cos(θ) sin(θ) is a product of these functions, and understanding their derivatives requires applying the product rule. Familiarity with the properties and derivatives of sine and cosine is crucial for solving the problem.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if u and v are functions of θ, then the derivative of their product is given by u'v + uv'. This rule is particularly relevant for differentiating y = cos(θ) sin(θ), as it allows for the correct application of calculus to find the first and subsequently the second derivative.

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Find y'' for the following functions.y = cos θ sin θ

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Key Concepts

Second Derivative

Trigonometric Functions

Product Rule

Find y'' for the following functions.
y = cos θ sin θ