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

The answer to (\partial)/(\partial x)(ln(x/y)) is 1/x

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

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

\frac{1}{x}

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{1}{x}

\int_{- a}^{a} e^{\frac{- x ^{2}}{a ^{2}}} d x

(- 4, 3), (3, 1)

\int \frac{x ^{2} - 15}{x + 7} d x

y^{^{''}} - 2 y^{^{'}} = 9 x

\int_{0}^{2} (3 x^{2} - 2) d x

What is (\partial)/(\partial x)(ln(x/y)) ?
The answer to (\partial)/(\partial x)(ln(x/y)) is 1/x