Find $\dfrac{dy}{dx}$ if $(x^{2}+y^{2})^{2}=xy$.

$\dfrac{y-4x(x^{2}+y^{2})}{4y(x^{2}+y^{2})-x}$

CBSE Class 12 · 2009 · DifferentiationLong Answer

Find

\frac{d y}{d x}

(x^{2} + y^{2})^{2} = x y

Answer: $\frac{y - 4 x ( x ^{2} + y ^{2} )}{4 y ( x ^{2} + y ^{2} ) - x}$

Differentiate both sides w.r.t.

x

(chain rule on the left, product rule on the right):

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

Expand and collect

\frac{d y}{d x}

terms:

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

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

CBSE Class 12 Board · 2009 · Subjective