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

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

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

expand

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

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

Solution steps

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

Expand

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

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

Apply the fraction rule:

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

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

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

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

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

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

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

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

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

(2 sin (x) + 1)^{2}

\frac{( 7 s - 31 )}{( ( s + 3 ) ( s + 4 ) )}

\frac{2 + 1}{( 2 - 1 ) ^{5}}

\frac{s}{( s + 8 ) ^{2}}

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