What is (\partial)/(\partial x)(ln((xy)^z)) ?

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

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(\partial)/(\partial x)(ln((xy)^z))

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

\frac{z}{x}

Solution steps

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

Treat

y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{z}{x}

\frac{\partial}{\partial x} (x^{2} + w^{2})

x \to 0 - lim (ln (sin (x)))

n = 1 \sum \infty \frac{1}{n + 1}

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

y = (arctan (2 x))^{2}

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