Chapter 2: Q. 76 (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 $\frac{d y}{d x}$
${(3 y^{2} + 5 x y - 2)}^{4} = 2$

Short Answer

Expert verified

$\frac{d y}{d x} = \frac{- 5 y}{(5 x + 6 y)}$

Step by step solution

Step 1. Given Information:

Given equation: ${(3 y^{2} + 5 x y - 2)}^{4} = 2$

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

$\frac{d}{d x} {(3 y^{2} + 5 x y - 2)}^{4} = \frac{d}{d x} 2$

Using product and chain rule we get

localid="1648700033816" $4 {(3 y^{2} + 5 x y - 2)}^{3} \frac{d}{d x} (3 y^{2} + 5 x y - 2) = \frac{d}{d x} 2 4 {(3 y^{2} + 5 x y - 2)}^{3} (\frac{d}{d x} (3 y^{2}) + \frac{d}{d x} (5 x y) - \frac{d}{d x} (2)) = \frac{d}{d x} 2 4 {(3 y^{2} + 5 x y - 2)}^{3} (6 y \frac{d y}{d x} + (5 x \frac{d y}{d x} + 5 y) - 0) = 0 (5 x + 6 y) \frac{d y}{d x} + 5 y = 0 (5 x + 6 y) \frac{d y}{d x} = - 5 y \frac{d y}{d x} = \frac{- 5 y}{(5 x + 6 y)}$

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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 $\frac{d y}{d x}$
${(3 y^{2} + 5 x y - 2)}^{4} = 2$

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 dydx(3y2+5xy−2)4=2

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Most popular questions from this chapter

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Pure 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 $\frac{d y}{d x}$
${(3 y^{2} + 5 x y - 2)}^{4} = 2$