Textbook Question
9–61. Evaluate and simplify y'.
y = e^sin x+2x+1
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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.R.23
Evaluate and simplify y'.
y = ln |sec 3x|
Verified step by step guidance1
Identify the given equation: y' * y = ln |sec(3x)|. This is a differential equation where y' is the derivative of y with respect to x.
Recognize that this is a separable differential equation. We can separate the variables y and x to solve it.
Rewrite the equation by dividing both sides by y: y' = (ln |sec(3x)|) / y.
Integrate both sides: Integrate the left side with respect to y and the right side with respect to x. The left side becomes ∫(1/y) dy, and the right side becomes ∫ln |sec(3x)| dx.
Solve the integrals: The integral of 1/y with respect to y is ln|y|, and the integral of ln |sec(3x)| with respect to x will require integration techniques such as integration by parts. After integrating, solve for y to find the general solution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Natural Logarithm
The derivative of the natural logarithm function, ln(x), is given by 1/x. When dealing with the natural logarithm of a function, such as ln|u|, the derivative is found using the chain rule, resulting in (1/u) * (du/dx). This principle is essential for differentiating functions involving logarithms.
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Derivative of the Natural Logarithmic Function
Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y is composed of another function u, then the derivative dy/dx can be calculated as dy/du multiplied by du/dx. This rule is crucial when differentiating functions like ln|sec 3x|, where sec 3x is a composite function.
Recommended video:
05:02Intro to the Chain Rule
Trigonometric Derivatives
Understanding the derivatives of trigonometric functions is vital for calculus. For example, the derivative of sec(x) is sec(x)tan(x). In the context of the given function, knowing how to differentiate sec(3x) will be necessary to apply the chain rule effectively and find the derivative of y = ln|sec 3x|.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
9–61. Evaluate and simplify y'.
y = e^tan x (tan x−1)
Textbook Question
66–71. Higher-order derivatives Find and simplify y''.
x + sin y = y
1
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Textbook Question
9–61. Evaluate and simplify y'.
y = tan (sin θ)
Textbook Question
{Use of Tech} Beak length The length of the culmen (the upper ridge of a bird’s bill) of a t-week-old Indian spotted owlet is modeled by the function L(t)=11.94 / 1 + 4e^−1.65t, where L is measured in millimeters.
b. Use a graph of L′(t) to describe how the culmen grows over the first 5 weeks of life.
Textbook Question
Use the given graphs of f and g to find each derivative. <IMAGE>
c. d/dx ((f(x) / g(x)) |x=3