What is (\partial)/(\partial y)(x^5ln(y^5)) ?

The answer to (\partial)/(\partial y)(x^5ln(y^5)) is (5x^5)/y

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(\partial)/(\partial y)(x^5ln(y^5))

\frac{\partial}{\partial y} (x^{5} ln (y^{5}))

\frac{5 x ^{5}}{y}

Solution steps

\frac{\partial}{\partial y} (x^{5} ln (y^{5}))

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= x^{5} \frac{\partial}{\partial y} (ln (y^{5}))

Apply the chain rule:

\frac{1}{y ^{5}} \frac{\partial}{\partial y} (y^{5})

= \frac{1}{y ^{5}} \frac{\partial}{\partial y} (y^{5})

\frac{\partial}{\partial y} (y^{5}) = 5 y^{4}

= x^{5} \frac{1}{y ^{5}} \cdot 5 y^{4}

Simplify

x^{5} \frac{1}{y ^{5}} \cdot 5 y^{4} : \frac{5 x ^{5}}{y}

= \frac{5 x ^{5}}{y}

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What is (\partial)/(\partial y)(x^5ln(y^5)) ?
The answer to (\partial)/(\partial y)(x^5ln(y^5)) is (5x^5)/y