What is (\partial)/(\partial x)(ln(xy^2z^3)) ?

The answer to (\partial)/(\partial x)(ln(xy^2z^3)) is 1/x

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(\partial)/(\partial x)(ln(xy^2z^3))

\frac{\partial}{\partial x} (ln (x y^{2} z^{3}))

\frac{1}{x}

Solution steps

\frac{\partial}{\partial x} (ln (x y^{2} z^{3}))

Treat

y, z

as constants

Apply the chain rule:

\frac{1}{x y ^{2} z ^{3}} \frac{\partial}{\partial x} (x y^{2} z^{3})

= \frac{1}{x y ^{2} z ^{3}} \frac{\partial}{\partial x} (x y^{2} z^{3})

\frac{\partial}{\partial x} (x y^{2} z^{3}) = y^{2} z^{3}

= \frac{1}{x y ^{2} z ^{3}} y^{2} z^{3}

Simplify

\frac{1}{x y ^{2} z ^{3}} y^{2} z^{3} : \frac{1}{x}

= \frac{1}{x}

What is (\partial)/(\partial x)(ln(xy^2z^3)) ?
The answer to (\partial)/(\partial x)(ln(xy^2z^3)) is 1/x