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

The answer to (\partial)/(\partial y)(ln(x^5+y^5)) is (5y^4)/(x^5+y^5)

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

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

\frac{5 y ^{4}}{x ^{5} + y ^{5}}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{5 y ^{4}}{x ^{5} + y ^{5}}

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What is (\partial)/(\partial y)(ln(x^5+y^5)) ?
The answer to (\partial)/(\partial y)(ln(x^5+y^5)) is (5y^4)/(x^5+y^5)