Chapter 2: Q. 74 (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 $\frac{d y}{d x}$
role="math" localid="1648660297305" $x^{2} y - y^{2} x = x^{2} + 3$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information:

Given equation: $x^{2} y - y^{2} x = x^{2} + 3$

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

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

Using product and chain rule we get

role="math" localid="1648660863023" $x^{2} \frac{d}{d x} (y) + y \frac{d}{d x} (x^{2}) - (y^{2} \frac{d}{d x} (x) + x \frac{d}{d x} y^{2}) = \frac{d}{d x} (x^{2}) + \frac{d}{d x} (3) x^{2} \frac{d y}{d x} + y (2 x) - (y^{2} (1) + x (2 y) \frac{d y}{d x}) = (2 x) + 0 x^{2} \frac{d y}{d x} + 2 x y - y^{2} - 2 x y \frac{d y}{d x} = 2 x (x^{2} - 2 x y) \frac{d y}{d x} = 2 x - 2 x y + y^{2} \frac{d y}{d x} = \frac{2 x - 2 x y + y^{2}}{(x^{2} - 2 x y)}$

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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 $\frac{d y}{d x}$
role="math" localid="1648660297305" $x^{2} y - y^{2} x = x^{2} + 3$

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 dydxrole="math" localid="1648660297305" x2y−y2x=x2+3

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Decision Maths

Pure Maths

Geometry

Statistics

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 $\frac{d y}{d x}$
role="math" localid="1648660297305" $x^{2} y - y^{2} x = x^{2} + 3$