What is the derivative of (xy/(x^2-y^2)) ?

The derivative of (xy/(x^2-y^2)) is (x^3\frac{dy)/(dx)-x^2y+xy^2(dy)/(dx)-y^3}{(x^2-y^2)^2}

What is the first derivative of (xy/(x^2-y^2)) ?

The first derivative of (xy/(x^2-y^2)) is (x^3\frac{dy)/(dx)-x^2y+xy^2(dy)/(dx)-y^3}{(x^2-y^2)^2}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

\frac{x ^{3} \frac{d y}{d x} - x ^{2} y + x y ^{2} \frac{d y}{d x} - y ^{3}}{( x ^{2} - y ^{2} ) ^{2}}

Solution steps

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

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

Expand

(y + x \frac{d y}{d x}) (x^{2} - y^{2}) - (2 x - 2 y \frac{d y}{d x}) x y : x^{3} \frac{d y}{d x} - x^{2} y + x y^{2} \frac{d y}{d x} - y^{3}

= \frac{x ^{3} \frac{d y}{d x} - x ^{2} y + x y ^{2} \frac{d y}{d x} - y ^{3}}{( x ^{2} - y ^{2} ) ^{2}}

What is the derivative of (xy/(x^2-y^2)) ?
The derivative of (xy/(x^2-y^2)) is (x^3\frac{dy)/(dx)-x^2y+xy^2(dy)/(dx)-y^3}{(x^2-y^2)^2}
What is the first derivative of (xy/(x^2-y^2)) ?
The first derivative of (xy/(x^2-y^2)) is (x^3\frac{dy)/(dx)-x^2y+xy^2(dy)/(dx)-y^3}{(x^2-y^2)^2}