Ch. 3 - Derivatives

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

Hass 15th Edition

Ch. 3 - Derivatives

Problem 3.7.54

Chapter 3, Problem 3.7.54

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

Verified step by step guidance

Start by differentiating the given equation \(x^3 + y^2 = \sin^2(y)\) with respect to \(x\). Remember to apply implicit differentiation since \(y\) is a function of \(x\).

Differentiate \(x^3\) with respect to \(x\) to get \(3x^2\).

Differentiate \(y^2\) with respect to \(x\) using the chain rule: \(2y \cdot \frac{dy}{dx}\).

Differentiate \(\sin^2(y)\) with respect to \(x\) using the chain rule: \(2\sin(y)\cos(y) \cdot \frac{dy}{dx}\).

Set up the equation from the derivatives: \(3x^2 + 2y \cdot \frac{dy}{dx} = 2\sin(y)\cos(y) \cdot \frac{dy}{dx}\). Solve for \(\frac{dy}{dx}\) and then find \(\frac{dx}{dy}\) by taking the reciprocal of \(\frac{dy}{dx}\).

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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 find the derivative of a function when it is not explicitly solved for one variable in terms of another. In the equation x³ + y² = sin²y, both x and y are mixed together, requiring implicit differentiation to find dy/dx by differentiating both sides with respect to x, treating y as a function of x.

Inverse Function Theorem

The inverse function theorem provides a relationship between the derivatives of inverse functions. If dy/dx is the derivative of y with respect to x, then dx/dy is the reciprocal of dy/dx, assuming both derivatives exist and are non-zero. This theorem helps understand how dy/dx and dx/dy are related, as they are inverses of each other.

Chain Rule

The chain rule is essential for differentiating composite functions. When finding dy/dx or dx/dy, the chain rule allows us to differentiate expressions involving y as a function of x or x as a function of y. For example, differentiating sin²y with respect to x involves using the chain rule to account for y being a function of x.

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Intro to the Chain Rule

Related Practice

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In Exercises 17–28, find dy.

y = (2√x)/(3(1 + √x))

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Textbook Question

[Technology Exercise]

Graph the curves in Exercises 39–48.

a. Where do the graphs appear to have vertical tangent lines?

b. Confirm your findings in part (a) with limit calculations. But before you do, read the introduction to Exercises 37 and 38.

y = 4x²/⁵ − 2x

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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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In Exercises 1–5, find the linearization L(x) of f(x) at x = a.

f(x) = ∛x, a = −8