What is (\partial)/(\partial y)(ln(x^{10}y)) ?

The answer to (\partial)/(\partial y)(ln(x^{10}y)) is 1/y

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(\partial)/(\partial y)(ln(x^{10}y))

\frac{\partial}{\partial y} (ln (x^{10} y))

\frac{1}{y}

Solution steps

\frac{\partial}{\partial y} (ln (x^{10} y))

Treat

x

as a constant

Apply the chain rule:

\frac{1}{x ^{10} y} \frac{\partial}{\partial y} (x^{10} y)

= \frac{1}{x ^{10} y} \frac{\partial}{\partial y} (x^{10} y)

\frac{\partial}{\partial y} (x^{10} y) = x^{10}

= \frac{1}{x ^{10} y} x^{10}

Simplify

\frac{1}{x ^{10} y} x^{10} : \frac{1}{y}

= \frac{1}{y}

What is (\partial)/(\partial y)(ln(x^{10}y)) ?
The answer to (\partial)/(\partial y)(ln(x^{10}y)) is 1/y