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)=(x+1)², x≥-1
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.13
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = (x^2 - 2x + 2)e^(x)
Verified step by step guidance1
Identify the function y = (x^2 - 2x + 2) e^{x} as a product of two functions: u = (x^2 - 2x + 2) and v = e^{x}.
Recall the product rule for derivatives: if y = u v, then \( \frac{dy}{dx} = u' v + u v' \).
Find the derivative of u with respect to x: \( u = x^2 - 2x + 2 \), so \( u' = 2x - 2 \).
Find the derivative of v with respect to x: \( v = e^{x} \), so \( v' = e^{x} \).
Apply the product rule: \( \frac{dy}{dx} = (2x - 2) e^{x} + (x^2 - 2x + 2) e^{x} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product Rule
The product rule is used to differentiate functions that are products of two or more functions. It states that the derivative of f(x)g(x) is f'(x)g(x) + f(x)g'(x). This rule is essential when differentiating y = (x^2 - 2x + 2)e^x, where both parts depend on x.
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The Product Rule
Derivative of Polynomial Functions
Polynomial functions like x^2 - 2x + 2 are differentiated term-by-term using the power rule. The power rule states that d/dx[x^n] = nx^(n-1). Understanding how to differentiate each term correctly is crucial for applying the product rule effectively.
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Derivative of Exponential Functions
The derivative of the exponential function e^x is e^x itself. This property simplifies differentiation when e^x is part of the function. Recognizing this helps in applying the product rule to functions involving e^x.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 53–76.
57. ∫dx/(x√(25x²-2))
Textbook Question
137. Find a curve through the origin in the xy-plane whose length from x = 0 to x = 1 is L = ∫ from 0 to 1 of sqrt(1 + (1/4)e^x) dx.
Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
126. eʸ = y^(ln x)
Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
25. y = ln(ln(x))
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Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
7. lim (x → 2) (x - 2) / (x² - 4)