Textbook Question
If f′(x)=3x+2, find the slope of the line tangent to the curve y=f(x) at x=1, 2, and 3.
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.58
Evaluate and simplify y'.
sin x cos(y−1) = 1/2
Verified step by step guidance1
First, identify the given equation: \( y' \cdot \sin(x) \cdot \cos(y - 1) = \frac{1}{2} \). Here, \( y' \) represents the derivative of \( y \) with respect to \( x \).
To isolate \( y' \), divide both sides of the equation by \( \sin(x) \cdot \cos(y - 1) \). This gives: \( y' = \frac{1}{2} \div (\sin(x) \cdot \cos(y - 1)) \).
Simplify the expression for \( y' \) by rewriting the division as a multiplication by the reciprocal: \( y' = \frac{1}{2} \cdot \frac{1}{\sin(x) \cdot \cos(y - 1)} \).
Combine the fractions to express \( y' \) as a single fraction: \( y' = \frac{1}{2 \cdot \sin(x) \cdot \cos(y - 1)} \).
The expression for \( y' \) is now simplified. Ensure that the domain restrictions for \( \sin(x) \) and \( \cos(y - 1) \) are considered, as they cannot be zero.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent variable is not isolated on one side. In this case, we have a relationship between x and y given by sin x cos(y−1) = 1/2. By differentiating both sides with respect to x, we can find dy/dx, which represents y'.
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Chain Rule
The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When differentiating terms involving y, such as cos(y−1), we apply the chain rule to account for the derivative of the inner function (y−1) with respect to x, which involves dy/dx. This is crucial for correctly finding y'.
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In this problem, understanding the properties of sine and cosine is essential, especially since we are working with sin x and cos(y−1). These identities can help simplify the expression after differentiation and aid in solving for y'.
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Related Practice
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