Textbook Question
Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(sec⁴x tan² x)
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 3, Problem 3.5.64
Find y'' for the following functions.
y = cos θ sin θ
Verified step by step guidance1
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.
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The Second Derivative Test: Finding Local Extrema
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.
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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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Related Practice
Textbook Question
Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
lim x→π/2 cos x/x−π/2
Textbook Question
A cost function of the form C(x) = 1/2x² reflects diminishing returns to scale. Find and graph the cost, average cost, and marginal cost functions. Interpret the graphs and explain the idea of diminishing returns.
Textbook Question
Evaluate the derivative of the following functions.
f(x) = sin-1 (e-2x)
Textbook Question
Find the derivative of the following functions.
y = In √x⁴+x²
Textbook Question
Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.
(lim h🠂0) (2+h)⁴-16 / h