Textbook Question
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
73. lim (x → ∞) (2^x - 3^x) / (3^x + 4^x)
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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 7.3.9
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(5-7x)
Verified step by step guidance1
Identify the function given: \(y = e^{5 - 7x}\). This is an exponential function where the exponent is a linear expression in \(x\).
Recall the chain rule for differentiation: if \(y = e^{u(x)}\), then \(\frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx}\), where \(u(x)\) is the exponent function.
Set \(u(x) = 5 - 7x\). Next, find the derivative of \(u(x)\) with respect to \(x\): \(\frac{du}{dx} = -7\).
Apply the chain rule by multiplying the original function by the derivative of the exponent: \(\frac{dy}{dx} = e^{5 - 7x} \cdot (-7)\).
Write the final expression for the derivative as \(\frac{dy}{dx} = -7 e^{5 - 7x}\) (do not simplify further if not required).
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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^u, is found by multiplying e^u by the derivative of the exponent u. This uses the chain rule and reflects how the rate of change depends on both the function and its exponent.
Recommended video:
04:50
Derivatives of General Exponential Functions
Chain Rule
The chain rule is a method for differentiating composite functions. It states that the derivative of f(g(x)) is f'(g(x)) times g'(x). This is essential when the exponent itself is a function of x, like 5 - 7x in this problem.
Recommended video:
05:02Intro to the Chain Rule
Basic Differentiation Rules
Understanding how to differentiate constants and linear functions is fundamental. For example, the derivative of a constant is zero, and the derivative of -7x is -7. These rules help simplify the derivative of the exponent in the given function.
Recommended video:
04:00Solutions to Basic Differential Equations
Related Practice
Textbook Question
Evaluate the integrals in Exercises 33–54.
∫ (e^(√r) / √r) dr
Textbook Question
Show that the graph of the inverse of f(x)=mx+b, where m and b are constants and m≠0, is a line with slope 1/m and y-intercept -b/m.
Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
29. y = (1 - t)coth⁻¹(√t)
Textbook Question
Evaluate the integrals in Exercises 33–54.
∫(from ln3 to ln2) (e^x) dx
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
77. y = log₃(((x + 1)/(x − 1))^(ln 3))