What is (\partial)/(\partial x)(ln(x^y)) ?

The answer to (\partial)/(\partial x)(ln(x^y)) is y/x

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(\partial)/(\partial x)(ln(x^y))

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

\frac{y}{x}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{x ^{y}} y x^{y - 1} : \frac{y}{x}

= \frac{y}{x}

x \to 0 lim (\frac{1}{x} - \frac{1}{e ^{t} - 1})

x \to 4 lim (\frac{3 x + 4}{x - 2})

\frac{d}{d x} (\frac{x - 1}{x})

\frac{d}{d x} (\frac{a}{x ^{3}})

\int \frac{1}{cos ^{2} ( u )} d u

What is (\partial)/(\partial x)(ln(x^y)) ?
The answer to (\partial)/(\partial x)(ln(x^y)) is y/x