What is expand ((y-x)(y+x)(xy'(x)-y))/(x^2y^2) ?

The solution to expand ((y-x)(y+x)(xy'(x)-y))/(x^2y^2) is (y'(x))/x-y/(x^2)-(xy'(x))/(y^2)+1/y

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expand ((y-x)(y+x)(xy'(x)-y))/(x^2y^2)

expand

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

\frac{y ' ( x )}{x} - \frac{y}{x ^{2}} - \frac{x y ' ( x )}{y ^{2}} + \frac{1}{y}

Solution steps

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

Expand

(y - x) (y + x) (x y^{'} (x) - y) : x y^{2} y^{'} (x) - y^{3} - x^{3} y^{'} (x) + x^{2} y

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

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

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

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

\frac{x y ^{2} y ^{'} ( x )}{x ^{2} y ^{2}} = \frac{y ^{'} ( x )}{x}

\frac{y ^{3}}{x ^{2} y ^{2}} = \frac{y}{x ^{2}}

\frac{x ^{3} y ^{'} ( x )}{x ^{2} y ^{2}} = \frac{x y ^{'} ( x )}{y ^{2}}

\frac{x ^{2} y}{x ^{2} y ^{2}} = \frac{1}{y}

= \frac{y ^{^{'}} ( x )}{x} - \frac{y}{x ^{2}} - \frac{x y ^{^{'}} ( x )}{y ^{2}} + \frac{1}{y}

(x + 1) e^{2}

\frac{6}{s ( s + 2 ) ^{2} ( s + 4 )}

21.36204 (\frac{y ^{5}}{( 2 y + 3 ) ^{2}})^{\frac{1}{3}}

\frac{3 x + 11}{( x - 3 ) ( x + 2 )}

What is expand ((y-x)(y+x)(xy'(x)-y))/(x^2y^2) ?
The solution to expand ((y-x)(y+x)(xy'(x)-y))/(x^2y^2) is (y'(x))/x-y/(x^2)-(xy'(x))/(y^2)+1/y