What is (\partial)/(\partial x)((9x^2)/((x-y)^2)) ?

The answer to (\partial)/(\partial x)((9x^2)/((x-y)^2)) is -(18xy)/((x-y)^3)

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(\partial)/(\partial x)((9x^2)/((x-y)^2))

\frac{\partial}{\partial x} (\frac{9 x ^{2}}{( x - y ) ^{2}})

- \frac{18 x y}{( x - y ) ^{3}}

Solution steps

\frac{\partial}{\partial x} (\frac{9 x ^{2}}{( x - y ) ^{2}})

Treat

y

as a constant

Take the constant out:

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

= 9 \frac{\partial}{\partial x} (\frac{x ^{2}}{( x - y ) ^{2}})

Apply the Quotient Rule:

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

= 9 \cdot \frac{\frac{\partial}{\partial x} ( x ^{2} ) ( x - y ) ^{2} - \frac{\partial}{\partial x} ( ( x - y ) ^{2} ) x ^{2}}{( ( x - y ) ^{2} ) ^{2}}

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

= 9 \cdot \frac{2 x ( x - y ) ^{2} - 2 ( x - y ) x ^{2}}{( ( x - y ) ^{2} ) ^{2}}

Simplify

9 \cdot \frac{2 x ( x - y ) ^{2} - 2 ( x - y ) x ^{2}}{( ( x - y ) ^{2} ) ^{2}} : - \frac{18 x y}{( x - y ) ^{3}}

= - \frac{18 x y}{( x - y ) ^{3}}

\int_{0}^{1} (x) d x

\int - \frac{3 ( x + a )}{x ^{2} + a ^{2}} d x

\int cot^{6} (2 x) d x

\frac{d y}{d x}, y = (arctan (9 x))^{2}

\int_{0}^{\infty} \frac{1}{x ^{2} + 9} d x

What is (\partial)/(\partial x)((9x^2)/((x-y)^2)) ?
The answer to (\partial)/(\partial x)((9x^2)/((x-y)^2)) is -(18xy)/((x-y)^3)