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

The answer to (\partial)/(\partial y)(ln(x+sqrt(y))) is 1/(2sqrt(y)(x+\sqrt{y))}

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(\partial)/(\partial y)(ln(x+sqrt(y)))

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

\frac{1}{2 y ( x + y )}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

\frac{\partial}{\partial y} (x + y) = \frac{1}{2 y}

= \frac{1}{x + y} \cdot \frac{1}{2 y}

Simplify

\frac{1}{x + y} \cdot \frac{1}{2 y} : \frac{1}{2 y ( x + y )}

= \frac{1}{2 y ( x + y )}

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

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What is (\partial)/(\partial y)(ln(x+sqrt(y))) ?
The answer to (\partial)/(\partial y)(ln(x+sqrt(y))) is 1/(2sqrt(y)(x+\sqrt{y))}