Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
75. y = x³ log₁₀ x
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 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 7.3.127
For Exercises 127 and 128 find a function f satisfying each equation.
127. ∫₂ˣ √(f(t)) dt = x ln x
Verified step by step guidance1
Recognize that the problem gives an integral equation: \(\int_{2}^{x} \sqrt{f(t)} \, dt = x \ln x\). Our goal is to find the function \(f(x)\) that satisfies this equation.
Apply the Fundamental Theorem of Calculus by differentiating both sides of the equation with respect to \(x\). This gives: \(\frac{d}{dx} \int_{2}^{x} \sqrt{f(t)} \, dt = \frac{d}{dx} (x \ln x)\).
By the Fundamental Theorem of Calculus, the derivative of the integral with variable upper limit is the integrand evaluated at \(x\): \(\sqrt{f(x)} = \frac{d}{dx} (x \ln x)\).
Compute the derivative on the right side using the product rule: \(\frac{d}{dx} (x \ln x) = \ln x + 1\).
Set \(\sqrt{f(x)} = \ln x + 1\) and solve for \(f(x)\) by squaring both sides: \(f(x) = (\ln x + 1)^2\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Fundamental Theorem of Calculus
This theorem links differentiation and integration, stating that if F(x) is defined as an integral of f(t) from a constant to x, then F'(x) = f(x). It allows us to find the original function inside an integral by differentiating the integral expression.
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Fundamental Theorem of Calculus Part 1
Differentiation of Functions Involving Logarithms
Understanding how to differentiate functions like x ln x is essential. Using the product rule, the derivative of x ln x is ln x + 1, which helps in equating the derivative of the integral to the function inside the integral.
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06:30Logarithmic Differentiation
Solving for the Unknown Function Inside an Integral
Given an integral equation, we use differentiation to isolate and solve for the unknown function f(t). This involves applying the Fundamental Theorem of Calculus and algebraic manipulation to express f(x) explicitly.
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05:11Integrals of General Exponential Functions
Related Practice
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