Question

y=(x² - y²)² = 4xy
find derivative of y w.r.t. x​

Answer 1

${\huge{\underline{\purple{\bf\tt{Question:} } }}}$

y=(x² - y²)² = 4xy

Find derivative of y w.r.t. x

${\huge{\underline{\orange{\bf\tt{Solution:} } }}}$

$\sf (x^2-y^2)^2=4xy \\\\\sf x^4+y^4-2x^2y^2=4xy \\\\\sf Now, ~\frac{dy}{dx} \\\\\sf\implies \frac{d(x^4+y^4-2x^2y^2)} {dx} =\frac{d(4xy)}{dx} \\\\\sf\implies 4x^3-2[2xy^2+x^22y \frac{dy}{dx}] +4y^3\frac{dy}{dx}=4x\frac{dy}{dx}+4y \\\\\sf \implies4x^3-4xy^2-4x^2y \frac{dy}{dx} +4y^3\frac{dy}{dx}=4x\frac{dy}{dx}+4y \\\\\sf \implies 4x^3-4xy^2-4y=4x\frac{dy}{dx}+4x^2y \frac{dy}{dx} - 4y^3\frac{dy}{dx} \\\\\sf\implies 4x^3-4xy^2-4y=(4x+4x^2y-4y^3)\frac{dy}{dx} \\\\\sf\implies \frac{dy}{dx}=\frac{4x^3-4xy^2-4y}{4x+4x^2y-4y^3} \\\\\sf\implies \frac{x^3-xy^2-y}{x+x^2y-y^3}$