Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
71. y = log₂(5θ)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.11
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = xe^x-e^x
Verified step by step guidance1
Identify the function given: \(y = xe^x - e^x\).
Recognize that the function is composed of two terms: \(xe^x\) and \(-e^x\), and you will need to differentiate each term separately.
For the first term \(xe^x\), apply the product rule for differentiation, which states: \(\frac{d}{dx}[u v] = u' v + u v'\). Here, let \(u = x\) and \(v = e^x\).
Calculate the derivatives of \(u\) and \(v\): \(u' = \frac{d}{dx}[x] = 1\) and \(v' = \frac{d}{dx}[e^x] = e^x\).
Apply the product rule to the first term: \(\frac{d}{dx}[xe^x] = 1 \cdot e^x + x \cdot e^x = e^x + xe^x\). Then differentiate the second term \(-e^x\) as \(-e^x\). Finally, combine these results to write the derivative of \(y\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Exponential Functions
The derivative of the exponential function e^x is e^x itself. This property is fundamental when differentiating expressions involving e^x, as it simplifies the process and helps in applying rules like the product and chain rules effectively.
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Derivatives of General Exponential Functions
Product Rule
The product rule is used to differentiate functions that are products of two differentiable functions. It states that the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x). This rule is essential for differentiating terms like xe^x.
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05:18The Product Rule
Basic Differentiation Rules
Basic differentiation rules include the power rule, constant multiple rule, and sum/difference rule. These rules allow us to differentiate terms like x and combine derivatives of multiple terms, such as in y = xe^x - e^x.
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04:00Solutions to Basic Differential Equations
Related Practice
Textbook Question
Evaluate the integrals in Exercises 39–56.
45. ∫(from 1 to 2)(2ln x)/x dx
1
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Textbook Question
Evaluate the integrals in Exercises 87–96.
89. ∫₀¹ 2^(−θ) dθ
Textbook Question
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x²+1, x≥0
Textbook Question
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?
Textbook Question
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
63. y = x^π"