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

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

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

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

\frac{1}{2 x}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{1}{2 x}

x \to 0 lim (e^{- \frac{1}{x ^{2}}})

\frac{6}{2 - 4 x}

\frac{d}{d x} (x e^{x^{2} y})

7 x^{2} y^{^{'}} = y^{^{'}} + 5 x e^{- y}

y^{^{''}} - 6 y^{^{'}} + 9 y = t^{- 3} e^{3 t}

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