What is (\partial)/(\partial y)(ln(x^3y)) ?

The answer to (\partial)/(\partial y)(ln(x^3y)) is 1/y

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(\partial)/(\partial y)(ln(x^3y))

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

\frac{1}{y}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{1}{y}

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