Textbook Question
Evaluate the integrals in Exercises 53–76.
55. ∫dx/(17+x²)
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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 7.2.12
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
12. y = ln(10/x)
Verified step by step guidance1
Recognize that the function is given as \(y = \ln\left(\frac{10}{x}\right)\), which is a natural logarithm of a quotient.
Use the logarithm property that \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\) to rewrite the function as \(y = \ln(10) - \ln(x)\).
Note that \(\ln(10)\) is a constant since 10 is a constant, so its derivative with respect to \(x\) is zero.
Recall the derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\), so the derivative of \(-\ln(x)\) is \(-\frac{1}{x}\).
Combine these results to write the derivative \(\frac{dy}{dx} = 0 - \frac{1}{x} = -\frac{1}{x}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of the Natural Logarithm Function
The derivative of ln(u), where u is a differentiable function of x, is (1/u) times the derivative of u. This rule allows us to differentiate logarithmic functions by applying the chain rule to the inner function.
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Derivative of the Natural Logarithmic Function
Chain Rule
The chain rule is used to differentiate composite functions. It states that the derivative of a function composed of another function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
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05:02Intro to the Chain Rule
Derivative of a Quotient
When differentiating a function expressed as a quotient, such as 10/x, it can be rewritten as 10x⁻¹. Differentiating powers of x involves applying the power rule, which states that d/dx of x^n is n*x^(n-1).
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06:43The Quotient Rule
Related Practice
Textbook Question
Evaluate the integrals in Exercises 87–96.
87. ∫ 5ˣ dx
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Textbook Question
Evaluate the integrals in Exercises 39–56.
43. ∫(from 0 to π)(sin t)/(2 - cos t) dt
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Textbook Question
Suppose that the range of g lies in the domain of f so that the composition fog is defined. If f and g are one-to-one, can anything be said about fog? Give reasons for your answer.
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Textbook Question
135. Find the area of the “triangular” region in the first quadrant that is bounded above by the curve y = e^(2x), below by the curve y = e^x, and on the right by the line x = ln(3).
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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
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