Textbook Question
Using identities Use the identity sin 2x=2 sin x cos x sin 2 to find d/dx (sin 2x). Then use the identity cos 2x = cos² x−sin² x to express the derivative of sin 2x in terms of cos 2x.
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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 3.4.21
Derivatives Find and simplify the derivative of the following functions.
f(x) = x /x+1
Verified step by step guidance1
Step 1: Recognize that the function \( f(x) = \frac{x}{x+1} \) is a quotient of two functions, \( u(x) = x \) and \( v(x) = x+1 \). To find the derivative, we will use the Quotient Rule.
Step 2: Recall the Quotient Rule, which states that if \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).
Step 3: Differentiate \( u(x) = x \) to get \( u'(x) = 1 \), and differentiate \( v(x) = x+1 \) to get \( v'(x) = 1 \).
Step 4: Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the Quotient Rule formula: \( f'(x) = \frac{1 \cdot (x+1) - x \cdot 1}{(x+1)^2} \).
Step 5: Simplify the expression obtained in Step 4 to find the simplified form of the derivative.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, quotient rule, and chain rule.
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Derivatives
Quotient Rule
The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If you have a function f(x) = g(x)/h(x), the derivative f'(x) is given by (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential when differentiating functions that are expressed as fractions.
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Simplification of Derivatives
After finding the derivative of a function, simplification is often necessary to express the result in its simplest form. This may involve factoring, reducing fractions, or combining like terms. Simplifying the derivative can make it easier to analyze the function's behavior, such as identifying critical points and determining concavity.
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Related Practice
Textbook Question
15–48. Derivatives Find the derivative of the following functions.
y = 10^x(In 10^x-1)
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Derivatives Find and simplify the derivative of the following functions.
f(t) = t⁵/³e^t
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Find y'' for the following functions.
y = ex sin x
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Evaluate the derivative of the following functions.
f(x) = sin-1 2x
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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(3x + 1)⁴
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