Textbook Question
Find the derivative of the following functions.
y = x sin 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 3.26
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
f(v) = v¹⁰⁰+e^v+10
Verified step by step guidance1
Step 1: Identify the function components. The function f(v) = v^{100} + e^v + 10 consists of three terms: v^{100}, e^v, and 10.
Step 2: Differentiate the first term v^{100}. Use the power rule, which states that the derivative of v^n is n*v^{n-1}. Therefore, the derivative of v^{100} is 100*v^{99}.
Step 3: Differentiate the second term e^v. The derivative of e^v with respect to v is simply e^v, as the exponential function is its own derivative.
Step 4: Differentiate the constant term 10. The derivative of a constant is 0, as constants do not change with respect to the variable.
Step 5: Combine the derivatives of each term. The derivative of the function f(v) is the sum of the derivatives of its individual terms: 100*v^{99} + e^v + 0.
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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 at which a function changes at any given point. It is a fundamental concept in calculus that measures how a function's output value changes as its input value changes. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.
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Power Rule
The Power Rule is a basic differentiation rule used to find the derivative of functions of the form f(x) = x^n, where n is a real number. According to this rule, the derivative is given by f'(x) = n*x^(n-1). This rule simplifies the process of differentiating polynomial functions, making it essential for solving problems involving powers of variables.
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Exponential Functions
Exponential functions are functions of the form f(x) = e^x, where e is the base of the natural logarithm. The derivative of an exponential function is unique because it is equal to the function itself, meaning f'(x) = e^x. Understanding how to differentiate exponential functions is crucial for solving calculus problems involving growth and decay models.
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Related Practice
Textbook Question
47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.
f(x)=4e^10x; (4,0)
Textbook Question
67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.
f(x) = e^3x+1
Textbook Question
Derivative calculations Evaluate the derivative of the following functions at the given point.
f(s) = 2√s-1; a=25
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Textbook Question
27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
6x³+7y³ = 13xy
Textbook Question
Find the slope of the line tangent to the graph of f(x) = x / x+6 at the point (3, 1/3) and at (-2, -1/2).