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

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

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

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

\frac{1}{2 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 x y} x

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

Simplify

\frac{1}{x y} \cdot \frac{1}{2 x y} x : \frac{1}{2 y}

= \frac{1}{2 y}

\int (3 y^{2} - 2 sin (y)) d y

(x - 2)^{2} (x - 3)^{3}

\int_{0}^{\infty} \frac{x}{x ^{3}} d x

(5 x^{6} + 2 x^{3})^{4}

f (x) = x e^{x} csc (x)

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