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

The answer to (\partial)/(\partial x)(x/(x^5-y^5)) is (-4x^5-y^5)/((x^5-y^5)^2)

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

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

\frac{- 4 x ^{5} - y ^{5}}{( x ^{5} - y ^{5} ) ^{2}}

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

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

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

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

\frac{\partial}{\partial x} (x^{5} - y^{5}) = 5 x^{4}

= \frac{1 \cdot ( x ^{5} - y ^{5} ) - 5 x ^{4} x}{( x ^{5} - y ^{5} ) ^{2}}

1 \cdot (x^{5} - y^{5}) - 5 x^{4} x = - 4 x^{5} - y^{5}

= \frac{- 4 x ^{5} - y ^{5}}{( x ^{5} - y ^{5} ) ^{2}}

What is (\partial)/(\partial x)(x/(x^5-y^5)) ?
The answer to (\partial)/(\partial x)(x/(x^5-y^5)) is (-4x^5-y^5)/((x^5-y^5)^2)