Chapter 2: Q. 73 (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 $\frac{d y}{d x}$
$(3 x + 1) (y^{2} - y + 6) = 0$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information:

Given equation: $(3 x + 1) (y^{2} - y + 6) = 0$

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

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

Using product and chain rule we get

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

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

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 dydx(3x+1)(y2−y+6)=0

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

Geometry

Theoretical and Mathematical Physics

Calculus

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Applied Mathematics

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