What is (\partial)/(\partial y)(ln(sqrt(1+xy))) ?

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

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(\partial)/(\partial y)(ln(sqrt(1+xy)))

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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