Use implicit differentiation to find dy/dx.
-$$\frac { x + y } { x - y } = x ^ { 2 } + y ^ { 2 }$$
A) $\frac { x ( x - y ) ^ { 2 } + y } { x + y ( x - y ) ^ { 2 } }$
B) $\frac { x ( x - y ) ^ { 2 } - y } { x - y ( x - y ) ^ { 2 } }$
C) $\frac { x ( x - y ) ^ { 2 } + y } { x - y ( x - y ) ^ { 2 } }$
D) $\frac { x ( x - y ) ^ { 2 } - y } { x + y ( x - y ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

We start by taking the derivative of both sides with respect to $x$ using the quotient rule and chain rule on the left side and the sum rule on the right side:

$$\frac{d}{dx}\left[\frac{x+y}{x-y}\right]=\frac{d}{dx}[x^2+y^2]$$

Simplifying the left side with the quotient rule:

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

$$\frac{(x-y)(1+\frac{dy}{dx})-(x+y)(1-\frac{dy}{dx})}{(x-y)^2}=2x+2y\frac{dy}{dx}$$

Expanding and simplifying:

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

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

Multiplying both sides by $(x-y)^2$ to eliminate the denominator:

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

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

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

$$2(x-y)\frac{dy}{dx}-2y=(x-y)^2(-y-y\frac{dy}{dx})$$

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

Arranging the terms and solving for $\frac{dy}{dx}$:

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

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

$$\frac{dy}{dx}=\frac{x(y-x)+(y+x)y}{(x-y)^2+y^2}=\frac{x^2+2xy+y^2}{(x-y)^2+y^2}=\frac{(x+y)^2}{(x-y)^2+y^2}$$

Simplifying this expression gives:

$$\frac{(x+y)^2}{(x-y)^2+y^2}=\frac{x^2+2xy+y^2}{x^2-2xy+y^2+y^2}=\frac{(x-y)^2+4xy}{(x-y)^2+2y^2}$$

We can further simplify by replacing $y^2$ with $x^2-x(y-x)$ (from the original equation) to get:

$$\frac{(x+y)^2}{(x-y)^2+y^2}=\frac{(x-y)^2+6xy-x(x-y)}{(x-y)^2+x(y-x)}=\frac{(x-y)^2+2xy}{(x-y)^2+x(y-x)}$$

This simplifies to:

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

So the answer is (C).

Use Implicit Differentiation to Find Dy/dx x+yx−y=x2+y2\frac { x + y } { x - y } = x ^ { 2 } + y ^ { 2 }x−yx+y​=x2+y2

Use Implicit Differentiation to Find Dy/dx $\frac { x + y } { x - y } = x ^ { 2 } + y ^ { 2 }$