Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
y = xe^y
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 3.5.23
Find the derivative of the following functions.
y = sin x + cos x
Verified step by step guidance1
Identify the function for which you need to find the derivative: \( y = \sin x + \cos x \).
Recall the basic derivatives of trigonometric functions: \( \frac{d}{dx}(\sin x) = \cos x \) and \( \frac{d}{dx}(\cos x) = -\sin x \).
Apply the sum rule for derivatives, which states that the derivative of a sum of functions is the sum of their derivatives.
Differentiate each term separately: \( \frac{d}{dx}(\sin x) = \cos x \) and \( \frac{d}{dx}(\cos x) = -\sin x \).
Combine the results to find the derivative of the entire function: \( \frac{dy}{dx} = \cos x - \sin x \).
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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. Derivatives are used to find rates of change and can be computed using various rules and techniques.
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Derivatives
Trigonometric Functions
Trigonometric functions, such as sine (sin) and cosine (cos), are fundamental functions in mathematics that relate angles to ratios of sides in right triangles. In calculus, these functions have specific derivatives: the derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Understanding these derivatives is essential for differentiating functions involving trigonometric expressions.
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Sum Rule of Derivatives
The sum rule of derivatives states that the derivative of a sum of functions is equal to the sum of their derivatives. This means that if you have a function that is the sum of two or more functions, you can differentiate each function separately and then add the results together. This rule simplifies the process of finding derivatives for functions like y = sin x + cos x.
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Related Practice
Textbook Question
23–51. Calculating derivatives Find the derivative of the following functions.
y = sin x / 1 + cos x
Textbook Question
If f is continuous at a, must f be differentiable at a?
Textbook Question
A water heater that has the shape of a right cylindrical tank with a radius of 1 ft and a height of 4 ft is being drained. How fast is water draining out of the tank (in ft³/min) if the water level is dropping at 6 min/in?
Textbook Question
Use the graph of f(x)=|x| to find f′(x).
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Textbook Question
Find the slope of the graph of f(x) = 2 + xe^x at the point (0, 2).
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