Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
119. y = (sin x)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 7.4.3
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x
Verified step by step guidance1
Identify the given function: \(y = \frac{1}{x} \int_{1}^{x} \frac{e^{t}}{t} \, dt\).
Apply the product rule and the Fundamental Theorem of Calculus to find \(y'\). Since \(y\) is a quotient involving an integral, write \(y\) as \(y = \frac{1}{x} \cdot F(x)\) where \(F(x) = \int_{1}^{x} \frac{e^{t}}{t} \, dt\).
Differentiate \(y\) using the quotient rule or product rule: \(y' = \frac{d}{dx} \left( \frac{1}{x} F(x) \right) = -\frac{1}{x^{2}} F(x) + \frac{1}{x} F'(x)\).
Use the Fundamental Theorem of Calculus to find \(F'(x) = \frac{e^{x}}{x}\), then substitute back into the expression for \(y'\).
Substitute \(y\) and \(y'\) into the differential equation \(x^{2} y' + x y\) and simplify to verify that it equals \(e^{x}\).
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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 connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative is the integrand evaluated at that limit. It allows us to differentiate functions defined by integrals, which is essential for finding y' in the given problem.
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Fundamental Theorem of Calculus Part 1
Differentiation of Functions Defined by Integrals
When a function is given as an integral with variable limits, its derivative involves applying the Fundamental Theorem of Calculus and the product or chain rule if multiplied by other functions. Understanding this helps compute y' for y = (1/x) ∫(1 to x) (e^t / t) dt.
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Verification of Solutions to Differential Equations
To verify a function solves a differential equation, substitute the function and its derivative into the equation and simplify. If both sides match, the function is a solution. This process confirms that y and y' satisfy x²y' + xy = e^x.
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Related Practice
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