Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = ln(3te^(-t))
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 7.3.29
In Exercises 27–32, find dy/dx.
e^(2x)=sin(x+3y)
Verified step by step guidance1
Identify that the equation involves both \( x \) and \( y \) implicitly, so we will use implicit differentiation to find \( \frac{dy}{dx} \).
Differentiate both sides of the equation \( e^{2x} = \sin(x + 3y) \) with respect to \( x \). For the left side, use the chain rule: \( \frac{d}{dx} e^{2x} = 2e^{2x} \).
For the right side, apply the chain rule to \( \sin(x + 3y) \): \( \frac{d}{dx} \sin(x + 3y) = \cos(x + 3y) \cdot \frac{d}{dx}(x + 3y) \).
Differentiate \( x + 3y \) with respect to \( x \): \( \frac{d}{dx}(x + 3y) = 1 + 3 \frac{dy}{dx} \) because \( y \) is a function of \( x \).
Set up the equation: \( 2e^{2x} = \cos(x + 3y) (1 + 3 \frac{dy}{dx}) \). Then solve for \( \frac{dy}{dx} \) by isolating it on one side.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is used when y is defined implicitly as a function of x, rather than explicitly. It involves differentiating both sides of an equation with respect to x, treating y as a function of x and applying the chain rule to terms involving y.
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Finding The Implicit Derivative
Chain Rule
The chain rule is a method for differentiating composite functions. When differentiating expressions like sin(x + 3y), you differentiate the outer function (sin) and multiply by the derivative of the inner function (x + 3y), applying the chain rule carefully to account for y as a function of x.
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05:02Intro to the Chain Rule
Solving for dy/dx
After differentiating implicitly, you collect all terms involving dy/dx on one side of the equation. Then, you solve algebraically for dy/dx to find the derivative of y with respect to x, which expresses the rate of change of y in terms of x and y.
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06:06Solving Separable Differential Equations
Related Practice
Textbook Question
For problems 49–52 use implicit differentiation to find dy/dx at the given point P.
51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)
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Textbook Question
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Find the limits in Exercises 103–110.
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Textbook Question
In Exercises 1–4, solve for t.
e^(sqrt(t)) = x^2
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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
17. lim (θ → π/2) (2θ - π) / cos(2π - θ)
Textbook Question
Solve the differential equation in Exercises 9–22.
19. y²(dy/dx) = 3x²y³ - 6x²
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