Textbook Question
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
5. lim (x → 0) (1 - cos x) / x²
1
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 7.3.13
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: \( \frac{dy}{dx} = (2x - 2) e^{x} + (x^2 - 2x + 2) e^{x} \).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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.
Recommended video:
05:18
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.
Recommended video:
07:00Taylor Polynomials
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.
Recommended video:
04:50Derivatives of General Exponential Functions
Related Practice
Textbook Question
Evaluate the integrals in Exercises 53–76.
57. ∫dx/(x√(25x²-2))
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)=x³-1
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))
1
views
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
7. lim (x → 2) (x - 2) / (x² - 4)