Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
85. lim(x→1) (x² + 3x - 4)/(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.PE.3
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
3. y = (1/4)xe^(4x) - (1/16)e^(4x)
Verified step by step guidance1
Identify the function to differentiate: \(y = \frac{1}{4} x e^{4x} - \frac{1}{16} e^{4x}\).
Recognize that the derivative of \(y\) with respect to \(x\) requires using the product rule for the first term \(\frac{1}{4} x e^{4x}\) and the chain rule for the exponential terms.
Apply the product rule to the first term: if \(u = \frac{1}{4} x\) and \(v = e^{4x}\), then \(\frac{d}{dx}(uv) = u'v + uv'\). Compute \(u' = \frac{1}{4}\) and \(v' = 4 e^{4x}\) using the chain rule.
Differentiate the second term \(- \frac{1}{16} e^{4x}\) using the chain rule: \(\frac{d}{dx} e^{4x} = 4 e^{4x}\), so multiply by the constant \(-\frac{1}{16}\).
Combine the derivatives from both terms to write the full expression for \(\frac{dy}{dx}\) before simplifying.
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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 an exponential function with base e, such as e^(kx), is found by applying the chain rule. Specifically, d/dx[e^(kx)] = k * e^(kx), where k is a constant. This rule is essential for differentiating terms like e^(4x).
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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. If y = u(x)v(x), then y' = u'v + uv'. This rule applies to terms like (1/4)x * e^(4x), where both factors depend on x.
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05:18The Product Rule
Constant Multiple Rule
The constant multiple rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. For example, d/dx[c * f(x)] = c * f'(x). This simplifies differentiation of terms like (1/4)xe^(4x) and (1/16)e^(4x).
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04:02The Power Rule
Related Practice
Textbook Question
133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
Textbook Question
Evaluate the integrals in Exercises 31–78.
41. ∫(from 0 to 4)2t/(t² - 25)dt
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Textbook Question
113. The function f(x) = e^x + x, being differentiable and one-to-one, has a differentiable inverse f^(-1)(x). Find the value of df^(-1)/dx at the point f (ln 2).
Textbook Question
109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
e. f(x) = arccsc(x), g(x) = 1/x
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(arccos(x))