Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.8.32

Chapter 3, Problem 3.8.32

Use implicit differentiation to find dy/dx.
e^xy = 2y

Verified step by step guidance

Start by differentiating both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms involving y, use implicit differentiation.

Differentiate the left side of the equation exy with respect to x. Use the product rule: d(uv)/dx = u'v + uv'. Here, u = e^x and v = y.

Differentiate e^x with respect to x, which is e^x, and differentiate y with respect to x, which is dy/dx. Apply the product rule: e^x * dy/dx + y * e^x.

Differentiate the right side of the equation 2y with respect to x. Since y is a function of x, the derivative is 2 * dy/dx.

Set the derivatives from both sides equal to each other: e^x * dy/dx + y * e^x = 2 * dy/dx. Solve for dy/dx by isolating it on one side of the equation.

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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 a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for equations involving products or powers of variables.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation when dealing with terms like e^(xy).

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a constant and 'x' is the variable. In the context of the given equation, e^(xy) represents an exponential function where the exponent is a product of two variables. Understanding the properties of exponential functions, such as their derivatives, is crucial for applying implicit differentiation effectively.

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Related Practice

Textbook Question

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 (sin 3x) / x

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Textbook Question

Derivatives Find the derivative of the following functions. See Example 2 of Section 3.2 for the derivative of √x.

y = x⁵

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Textbook Question

Use implicit differentiation to find dy/dx.

x³ = (x + y) / (x - y)

Textbook Question

Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.

r(t) = (e^2t+ 3e^t + 2) / (e^t + 2)

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Textbook Question

Calculate the derivative of the following functions (i) using the fact that b^x = e^{xIn b} and (ii) using logarithmic differentiation. Verify that both answers are the same.

y = (x²+1)^x

Textbook Question

Find the derivative of the following functions by first expanding or simplifying the expression. Simplify your answers.

y = (x² - 2ax + a²) / (x - a); a is a constant.

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Use implicit differentiation to find dy/dx.exy = 2y

Verified video answer for a similar problem:

Key Concepts

Implicit Differentiation

Chain Rule

Exponential Functions

Use implicit differentiation to find dy/dx.
e^xy = 2y