Chapter 2: Q. 78 (page 211)

Each of the equations in Exercises 69–80 defines y as an implicit function of x. Use implicit differentiation (without solving for y first) to find $\frac{d y}{d x}$
$3 y = 5 x^{2} + \sqrt[3]{y - 2}$

Short Answer

Expert verified

$\frac{d y}{d x} = \frac{30 x {(y - 2)}^{\frac{2}{3}}}{9 {(y - 2)}^{\frac{2}{3}} - 1}$

Step by step solution

Step 1. Given Information:

Given equation: $3 y = 5 x^{2} + \sqrt[3]{y - 2}$

We want to find $\frac{d y}{d x}$ defines y as an implicit function of x by use implicit differentiation.

Step 2. Solution:

Differentiate both sides w.r.t. x

$\frac{d}{d x} 3 y = \frac{d}{d x} (5 x^{2} + \sqrt[3]{y - 2}) \frac{d}{d x} 3 y = \frac{d}{d x} 5 x^{2} + \frac{d}{d x} \sqrt[3]{y - 2}$

Using product and chain rule we get

$3 \frac{d y}{d x} = 10 x + \frac{1}{3} {(y - 2)}^{\frac{1}{3} - 1} \frac{d}{d x} (y - 2) 3 \frac{d y}{d x} = 10 x + \frac{1}{3 {(y - 2)}^{\frac{2}{3}}} \frac{d y}{d x} 3 \frac{d y}{d x} - \frac{1}{3 {(y - 2)}^{\frac{2}{3}}} \frac{d y}{d x} = 10 x (3 - \frac{1}{3 {(y - 2)}^{\frac{2}{3}}}) \frac{d y}{d x} = 10 x (\frac{9 {(y - 2)}^{\frac{2}{3}} - 1}{3 {(y - 2)}^{\frac{2}{3}}}) \frac{d y}{d x} = 10 x \frac{d y}{d x} = \frac{30 x {(y - 2)}^{\frac{2}{3}}}{9 {(y - 2)}^{\frac{2}{3}} - 1}$

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Each of the equations in Exercises 69–80 defines y as an implicit function of x. Use implicit differentiation (without solving for y first) to find $\frac{d y}{d x}$
$3 y = 5 x^{2} + \sqrt[3]{y - 2}$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Each of the equations in Exercises 69–80 defines y as an implicit function of x. Use implicit differentiation (without solving for y first) to find dydx3y=5x2+y-23

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

Each of the equations in Exercises 69–80 defines y as an implicit function of x. Use implicit differentiation (without solving for y first) to find $\frac{d y}{d x}$
$3 y = 5 x^{2} + \sqrt[3]{y - 2}$