What is (\partial)/(\partial y)((5xy)/(10x-y)) ?

The answer to (\partial)/(\partial y)((5xy)/(10x-y)) is (50x^2)/((10x-y)^2)

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(\partial)/(\partial y)((5xy)/(10x-y))

\frac{\partial}{\partial y} (\frac{5 x y}{10 x - y})

\frac{50 x ^{2}}{( 10 x - y ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{5 x y}{10 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}{10 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 ) ( 10 x - y ) - \frac{\partial}{\partial y} ( 10 x - y ) y}{( 10 x - y ) ^{2}}

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

\frac{\partial}{\partial y} (10 x - y) = - 1

= 5 x \frac{1 \cdot ( 10 x - y ) - ( - 1 ) y}{( 10 x - y ) ^{2}}

Simplify

5 x \frac{1 \cdot ( 10 x - y ) - ( - 1 ) y}{( 10 x - y ) ^{2}} : \frac{50 x ^{2}}{( 10 x - y ) ^{2}}

= \frac{50 x ^{2}}{( 10 x - y ) ^{2}}

\int_{- 8}^{8} (64 - x^{2}) d x

\frac{2}{( s + 3 )}

\frac{d}{d x} (\frac{1}{sin ( 3 x )})

\frac{d}{d x} (\frac{x ^{2} + 2 x - 6}{( x + 1 ) ^{2}})

(x^{2} + 1) y^{^{''}} - 2 x y^{^{'}} + 2 y = 0

What is (\partial)/(\partial y)((5xy)/(10x-y)) ?
The answer to (\partial)/(\partial y)((5xy)/(10x-y)) is (50x^2)/((10x-y)^2)