Chapter 2: Q. 72 (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 $\frac{d y}{d x}$
$y^{6} - 3 x + 4 = 0$

Short Answer

Expert verified

$\frac{d y}{d x} = \frac{1}{2 y}$

Step by step solution

Step 1. Given Information:

Given equation: $y^{6} - 3 x + 4 = 0$

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

$\frac{d}{d x} (y^{6} - 3 x + 4) = \frac{d}{d x} 0 \frac{d}{d x} (y^{6}) - \frac{d}{d x} (3 x) + \frac{d}{d x} (4) = \frac{d}{d x} 0 6 y \cdot \frac{d y}{d x} - 3 + 0 = 0 6 y \cdot \frac{d y}{d x} = 3 \frac{d y}{d x} = \frac{3}{6 y} \frac{d y}{d x} = \frac{1}{2 y}$

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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 $\frac{d y}{d x}$
$y^{6} - 3 x + 4 = 0$

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 dydxy6-3x+4=0

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

One App. One Place for Learning.

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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 $\frac{d y}{d x}$
$y^{6} - 3 x + 4 = 0$