What is (\partial)/(\partial x)((-18x)/((x-y)^4)) ?

The answer to (\partial)/(\partial x)((-18x)/((x-y)^4)) is -(18(-3x-y))/((x-y)^5)

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(\partial)/(\partial x)((-18x)/((x-y)^4))

\frac{\partial}{\partial x} (\frac{- 18 x}{( x - y ) ^{4}})

- \frac{18 ( - 3 x - y )}{( x - y ) ^{5}}

Solution steps

\frac{\partial}{\partial x} (\frac{- 18 x}{( x - y ) ^{4}})

Simplify

\frac{- 18 x}{( x - y ) ^{4}} : - \frac{18 x}{( x - y ) ^{4}}

= \frac{\partial}{\partial x} (- \frac{18 x}{( x - y ) ^{4}})

Treat

y

as a constant

Take the constant out:

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

= - 18 \frac{\partial}{\partial x} (\frac{x}{( x - y ) ^{4}})

Apply the Quotient Rule:

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

= - 18 \cdot \frac{\frac{\partial}{\partial x} ( x ) ( x - y ) ^{4} - \frac{\partial}{\partial x} ( ( x - y ) ^{4} ) x}{( ( x - y ) ^{4} ) ^{2}}

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

\frac{\partial}{\partial x} ((x - y)^{4}) = 4 (x - y)^{3}

= - 18 \cdot \frac{1 \cdot ( x - y ) ^{4} - 4 ( x - y ) ^{3} x}{( ( x - y ) ^{4} ) ^{2}}

Simplify

- 18 \cdot \frac{1 \cdot ( x - y ) ^{4} - 4 ( x - y ) ^{3} x}{( ( x - y ) ^{4} ) ^{2}} : - \frac{18 ( - 3 x - y )}{( x - y ) ^{5}}

= - \frac{18 ( - 3 x - y )}{( x - y ) ^{5}}

What is (\partial)/(\partial x)((-18x)/((x-y)^4)) ?
The answer to (\partial)/(\partial x)((-18x)/((x-y)^4)) is -(18(-3x-y))/((x-y)^5)