Question

Differentiate implicitly to find \(d y / d x\). \(\frac{x}{x^{2}+y^{2}}-y^{2}=6\)

Short answer

So the derivative \(dy/dx\) of the given implicit function is: \(\frac{-x^3-y^3+3x^2y^2}{x^2+y^2+4xy^2}\)

Step-by-step solution

Differentiating each term

Firstly, differentiate each term of the equation with respect to x. The derivative of \(x\) with respect to \(x\) is \(1\), and for the term \(\frac{x}{x^{2}+y^{2}}\) we’ll have to use quotient rule \(\left(\frac{f}{g}\)'=\frac{f'g-g'f}{g^2}\). Differentiating \(y^{2}\) with respect to \(x\) will imply use of chain rule, resulting in \(2y(dy/dx)\). The derivative of the constant \(6\) with regards to \(x\) is \(0\)

Apply the quotient rule

The quotient rule states that the derivative of \(\frac{f}{g}\) is \(\frac{f'g-g'f}{g^2}\). We'll apply this to the term \(\frac{x}{x^{2}+y^{2}}\). This gives \(\frac{(1)(x^{2}+y^{2})-(2x + 2yy')x}{(x^{2}+y^{2})^{2}}\).

Simplify

After obtaining the derivatives of all terms, they are to be put back into the equation thereby giving: \(\frac{x^{2}+y^{2}-2x^{2}-2xyy'}{(x^{2}+y^{2})^{2}} -2y(dy/dx)=0\). Simplify and collect like terms.

Step 4. Solve for dy/dx

Rearrange the equation to solve for the derivative \(dy/dx\), and simplify to get the answer of: \(dy/dx = \frac{-x^3-y^3+3x^2y^2}{x^2+y^2+4xy^2}\)