What is (\partial)/(\partial x)(ln((y^6)/(x^{10)})) ?

The answer to (\partial)/(\partial x)(ln((y^6)/(x^{10)})) is -(10)/x

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(\partial)/(\partial x)(ln((y^6)/(x^{10)}))

\frac{\partial}{\partial x} (ln (\frac{y ^{6}}{x ^{10}}))

- \frac{10}{x}

Solution steps

\frac{\partial}{\partial x} (ln (\frac{y ^{6}}{x ^{10}}))

Treat

y

as a constant

Apply the chain rule:

\frac{1}{\frac{y ^{6}}{x ^{10}}} \frac{\partial}{\partial x} (\frac{y ^{6}}{x ^{10}})

= \frac{1}{\frac{y ^{6}}{x ^{10}}} \frac{\partial}{\partial x} (\frac{y ^{6}}{x ^{10}})

\frac{\partial}{\partial x} (\frac{y ^{6}}{x ^{10}}) = - \frac{10 y ^{6}}{x ^{11}}

= \frac{1}{\frac{y ^{6}}{x ^{10}}} (- \frac{10 y ^{6}}{x ^{11}})

Simplify

\frac{1}{\frac{y ^{6}}{x ^{10}}} (- \frac{10 y ^{6}}{x ^{11}}) : - \frac{10}{x}

= - \frac{10}{x}

\int 4 x^{2} e^{2 x} d x

\int 8 x e^{5 x} d x

x \to - \infty lim (2^{- x})

\frac{2 x}{9 x ^{2} + 1}

y = \frac{x}{x + 2}, (4, \frac{2}{6})

What is (\partial)/(\partial x)(ln((y^6)/(x^{10)})) ?
The answer to (\partial)/(\partial x)(ln((y^6)/(x^{10)})) is -(10)/x