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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.47
Chapter 3, Problem 3.47

In Exercises 43–50, find by implicit differentiation.
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√xy = 1

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1
Start by identifying the given equation: \( \sqrt{xy} = 1 \). This equation involves a square root, which can be rewritten for differentiation purposes.
Rewrite the equation to remove the square root: \( xy = 1^2 \) or simply \( xy = 1 \).
Differentiate both sides of the equation with respect to \( x \). Remember that \( y \) is a function of \( x \), so you'll need to use the product rule and implicit differentiation.
Apply the product rule to the left side: \( \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \). The right side differentiates to zero since it's a constant: \( \frac{d}{dx}(1) = 0 \).
Set the differentiated equation equal to zero: \( x \frac{dy}{dx} + y = 0 \). Solve for \( \frac{dy}{dx} \) to find the derivative of \( y \) with respect to \( x \).

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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 that are difficult or impossible to rearrange.
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Finding The Implicit Derivative

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 rule is essential when dealing with implicit differentiation, as it helps manage the derivatives of nested functions.
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Intro to the Chain Rule

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. It states that if u and v are functions of x, then the derivative of their product uv is given by u'v + uv'. This rule is particularly relevant in implicit differentiation when dealing with equations that involve products of variables, such as √xy, where both x and y are functions of the same variable.
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The Product Rule
Related Practice
Textbook Question

In Exercises 83–88, find equations for the lines that are tangent, and the lines that are normal, to the curve at the given point.


xy + 2x - 5y = 2, (3, 2)

Textbook Question

Derivatives in Differential Form


In Exercises 17–28, find dy.


y = 3 csc(1 − 2√x)

Textbook Question

Find the tangent line to the Witch of Agnesi (graphed here) at the point (2,1).

Textbook Question

The devil’s curve (Gabriel Cramer, 1750) Find the slopes of the devil’s curve y⁴ – 4y² = x⁴ – 9x² at the four indicated points.


Textbook Question

Implicit Differentiation


In Exercises 43–50, find by implicit differentiation.


xy + 2x + 3y = 1

Textbook Question

Second Derivatives


In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.


y² = x² + 2x