Textbook Question
Define the acceleration of an object moving in a straight line.
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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 3.19
Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.
y = x⁵
Verified step by step guidance1
Step 1: Identify the function for which you need to find the derivative. Here, the function is \( y = x^5 \).
Step 2: Recognize that this is a power function of the form \( y = x^n \), where \( n = 5 \).
Step 3: Apply the power rule for differentiation, which states that the derivative of \( x^n \) is \( nx^{n-1} \).
Step 4: Substitute \( n = 5 \) into the power rule formula to find the derivative: \( \frac{d}{dx}(x^5) = 5x^{5-1} \).
Step 5: Simplify the expression to obtain the derivative: \( 5x^4 \).
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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 a 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 graph of the function at a specific point.
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Derivatives
Power Rule
The Power Rule is a basic differentiation rule used to find the derivative of functions in the form of x^n, where n is a real number. According to this rule, the derivative of x^n is n*x^(n-1). This rule simplifies the process of differentiation for polynomial functions, making it easier to compute derivatives quickly.
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Function Notation
Function notation is a way to represent functions and their outputs. In the context of derivatives, it is common to express a function as y = f(x), where f(x) denotes the function's output for a given input x. Understanding function notation is essential for applying differentiation techniques and interpreting the results correctly.
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Related Practice
Textbook Question
How is lim x🠂0 sin x/x used in this section?
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Textbook Question
Use Theorem 3.10 to evaluate the following limits.
lim x🠂0 (sin 3x) / x
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Textbook Question
Use implicit differentiation to find dy/dx.
x3 = (x + y) / (x - y)
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Textbook Question
Calculate the derivative of the following functions (i) using the fact that bx = exIn b and (ii) using logarithmic differentiation. Verify that both answers are the same.
y = (x²+1)x
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Textbook Question
Match the graphs of the functions in a–d with the graphs of their derivatives in A–D. <MATCH A-D IMAGE>
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