What is (\partial)/(\partial x)(-(x-y)/(x^2(x+y))) ?

The answer to (\partial)/(\partial x)(-(x-y)/(x^2(x+y))) is -(2(-x^2+xy+y^2))/(x^3(x+y)^2)

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(\partial)/(\partial x)(-(x-y)/(x^2(x+y)))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Quotient Rule:

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

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

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

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

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

Simplify

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

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

f (x) = 2 (13 - x^{4})

\int_{1}^{8} \frac{1}{x 16 x ^{2} - 9} d x

x \to 6 lim (cos (\frac{1}{x - 6}))

\int cos (x) sec^{3} (x) d x

f (x) = \frac{x ^{3} + 3 x + 2}{x ^{2} - 1}

What is (\partial)/(\partial x)(-(x-y)/(x^2(x+y))) ?
The answer to (\partial)/(\partial x)(-(x-y)/(x^2(x+y))) is -(2(-x^2+xy+y^2))/(x^3(x+y)^2)