Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.9.22

Chapter 3, Problem 3.9.22

Derivatives in Differential Form

In Exercises 17–28, find dy.

xy² − 4x³/² − y = 0

Verified step by step guidance

First, identify the given equation: \( xy^2 - 4x^{3/2} - y = 0 \). This is an implicit function involving both x and y.

To find \( \frac{dy}{dx} \), we need to differentiate both sides of the equation with respect to x. Remember that y is a function of x, so use implicit differentiation.

Differentiate each term: For \( xy^2 \), use the product rule: \( \frac{d}{dx}(xy^2) = x \frac{d}{dx}(y^2) + y^2 \frac{d}{dx}(x) \). For \( y^2 \), use the chain rule: \( \frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \).

Differentiate \( -4x^{3/2} \) using the power rule: \( \frac{d}{dx}(-4x^{3/2}) = -4 \cdot \frac{3}{2}x^{1/2} \).

Differentiate \( -y \) with respect to x: \( \frac{d}{dx}(-y) = -\frac{dy}{dx} \). Combine all differentiated terms and solve for \( \frac{dy}{dx} \).

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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. In this method, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule when necessary. This allows us to find the derivative of one variable in terms of the other, even when they are intertwined.

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 composed of another function u, 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 crucial when dealing with implicit functions where one variable depends on another.

Differential Notation

Differential notation involves expressing derivatives in terms of differentials, such as dy and dx. In the context of implicit differentiation, dy represents the change in the dependent variable y, while dx represents the change in the independent variable x. Understanding this notation is essential for solving equations involving derivatives, as it helps clarify the relationship between the variables and their rates of change.

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Textbook Question

Derivatives in Differential Form

In Exercises 17–28, find dy.

y = (2√x)/(3(1 + √x))

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Textbook Question

Suppose that the function v in the Derivative Product Rule has a constant value c. What does the Derivative Product Rule then say? What does this say about the Derivative Constant Multiple Rule?

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Textbook Question

In Exercises 53 and 54, find both dy/dx (treating y as a differentiable function of x) and dx/dy (treating x as a differentiable function of y). How do dy/dx and dx/dy seem to be related?

54. x³ + y² = sin²y

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Textbook Question

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In Exercises 1–12, find the first and second derivatives.

y = x² + x + 8

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If L = √(x² + y²), dx/dt = –1, and dy/dt = 3, find dL/dt when x = 5 and y = 12.

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