Question

In Exercises \(1-8,\) find \(d y / d x\). $$x^{2}=\frac{x-y}{x+y}$$

Short answer

The derivative \(dy / dx\) for the given equation is \(y'=\frac{1-3x^2-2xy}{x^2+1}\)

Step-by-step solution

Distributing the \(x^{2}\)

Multiply both sides of the equation by \(x+y\) to get rid of the fraction: \(x^{2}(x+y) = x - y\)

Simplify the Equation

Open the bracket and simplify the equation to: \(x^{3} + x^{2} y = x - y\)

Rearrange the Equation

Rearrange equation to: \(x^{3} + x^{2} y -x + y = 0\)

Implicit Differentiation

Differentiate both sides with respect to \(x\) using implicit differentiation: \(\frac{d}{dx}(x^{3}) + \frac{d}{dx}(x^{2} y) - \frac{d}{dx}(x) + \frac{d}{dx}(y) = 0\) which simplifies to: \(3x^2 + 2xyy' + x^{2}y'-1+y' = 0\)

Solve for \(y'\)

From the previous step isolate \(y'\) on one side of the equation to find: \(y'=\frac{1-3x^2-2xy}{x^2+1}\)