What is (\partial)/(\partial x)((x/(x-y))^{1/3}) ?

The answer to (\partial)/(\partial x)((x/(x-y))^{1/3}) is -y/(3x^{2/3)(x-y)^{4/3}}

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(\partial)/(\partial x)((x/(x-y))^{1/3})

\frac{\partial}{\partial x} ((\frac{x}{x - y})^{\frac{1}{3}})

- \frac{y}{3 x ^{\frac{2}{3}} ( x - y ) ^{\frac{4}{3}}}

Solution steps

\frac{\partial}{\partial x} ((\frac{x}{x - y})^{\frac{1}{3}})

Treat

y

as a constant

Apply the chain rule:

\frac{1}{3 ( \frac{x}{x - y} ) ^{\frac{2}{3}}} \frac{\partial}{\partial x} (\frac{x}{x - y})

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

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

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

Simplify

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

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

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

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

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

3 x^{2} - 2 x

7 (4^{x})

What is (\partial)/(\partial x)((x/(x-y))^{1/3}) ?
The answer to (\partial)/(\partial x)((x/(x-y))^{1/3}) is -y/(3x^{2/3)(x-y)^{4/3}}