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

The answer to (\partial)/(\partial x)(ln((cos(x)y)^z)) is -ztan(x)

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

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

- z tan (x)

Solution steps

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

Treat

y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

= - z tan (x)

x \to 3 lim (3 x + 7)

\int_{0}^{2 π} cos (t) d t

x \to 5 lim (5 - x)

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

\int \frac{x ^{2}}{49 + 81 x ^{2}} d x

What is (\partial)/(\partial x)(ln((cos(x)y)^z)) ?
The answer to (\partial)/(\partial x)(ln((cos(x)y)^z)) is -ztan(x)