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

The answer to (\partial)/(\partial x)((y/x)^{1/z}) is -(y^{1/z}}{zx^{\frac{1+z)/z}}

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(\partial)/(\partial x)((y/x)^{1/z})

\frac{\partial}{\partial x} ((\frac{y}{x})^{\frac{1}{z}})

- \frac{y ^{\frac{1}{z}}}{z x ^{\frac{1 + z}{z}}}

Solution steps

\frac{\partial}{\partial x} ((\frac{y}{x})^{\frac{1}{z}})

Treat

y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

= - \frac{y ^{\frac{1}{z}}}{z x ^{\frac{1 + z}{z}}}

\frac{d}{d x} (r (x) (x)^{2} - x^{2})

\int_{2}^{4} (2 x + 2) d x

\int \frac{ln ( x )}{4 + x} d x

\int (x + e^{2 x}) d x

\int \frac{4}{b ^{4} x ^{2} + m ^{2}} d x

What is (\partial)/(\partial x)((y/x)^{1/z}) ?
The answer to (\partial)/(\partial x)((y/x)^{1/z}) is -(y^{1/z}}{zx^{\frac{1+z)/z}}