Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(sinh z)
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 7.3.7
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(-5x)
Verified step by step guidance1
Identify the function given: \(y = e^{-5x}\), where \(y\) is expressed in terms of \(x\).
Recall the chain rule for differentiation: if \(y = e^{u(x)}\), then \(\frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx}\).
Set the inner function \(u(x) = -5x\) and find its derivative with respect to \(x\): \(\frac{du}{dx} = -5\).
Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function: \(\frac{dy}{dx} = e^{-5x} \cdot (-5)\).
Write the final expression for the derivative as \(\frac{dy}{dx} = -5 e^{-5x}\) (do not calculate the numerical value).
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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 rule is essential for differentiating functions like y = e^(-5x), where the exponent is a function of x.
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Derivatives of General Exponential Functions
Chain Rule
The chain rule is used to differentiate composite functions. When a function is composed of an outer function and an inner function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For y = e^(-5x), the inner function is -5x.
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05:02Intro to the Chain Rule
Notation and Variable Dependence
Understanding which variable to differentiate with respect to (x, t, or θ) is crucial. In this problem, y is expressed in terms of x, so differentiation is with respect to x. Recognizing the independent variable ensures correct application of differentiation rules.
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04:22Sigma Notation
Related Practice
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Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
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Textbook Question
Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.
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