What is (\partial)/(\partial y)((5xy)/(x-y)) ?

The answer to (\partial)/(\partial y)((5xy)/(x-y)) is (5x^2)/((x-y)^2)

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(\partial)/(\partial y)((5xy)/(x-y))

\frac{\partial}{\partial y} (\frac{5 x y}{x - y})

\frac{5 x ^{2}}{( x - y ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{5 x y}{x - y})

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 5 x \frac{\partial}{\partial y} (\frac{y}{x - y})

Apply the Quotient Rule:

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

= 5 x \frac{\frac{\partial}{\partial y} ( y ) ( x - y ) - \frac{\partial}{\partial y} ( x - y ) y}{( x - y ) ^{2}}

\frac{\partial}{\partial y} (y) = 1

\frac{\partial}{\partial y} (x - y) = - 1

= 5 x \frac{1 \cdot ( x - y ) - ( - 1 ) y}{( x - y ) ^{2}}

Simplify

5 x \frac{1 \cdot ( x - y ) - ( - 1 ) y}{( x - y ) ^{2}} : \frac{5 x ^{2}}{( x - y ) ^{2}}

= \frac{5 x ^{2}}{( x - y ) ^{2}}

f (x) = x + \frac{3}{x}

(2 x - 1) e^{x}

y = 3 x^{2} - 4 x + 1

sin (x), 5

\frac{( 16 s + 5 )}{( s ^{2} + 9 )}

What is (\partial)/(\partial y)((5xy)/(x-y)) ?
The answer to (\partial)/(\partial y)((5xy)/(x-y)) is (5x^2)/((x-y)^2)