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

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

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

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

\frac{x}{y}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{x}{y}

t \to - 5 lim (t^{2} - 5)

y = 6 x sin (x), (\frac{π}{2}, 3 π)

\frac{e ^{- 7 s}}{( s - 3 ) ^{3}}

\int \frac{1}{x ^{4} ( x ^{3} + 1 ) ^{2}} d x

\int x \cdot x^{2} + 3 d x

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