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

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

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

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

\frac{1}{y}

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{1}{y}

x \to - 2 lim (4 - 3 x^{2})

x \to 0 + lim (3 + x^{8} ln (x))

\int - \frac{20}{y ^{21}} d y

f (x) = x^{2} - x^{4}, at x = 1

\frac{\partial}{\partial x} (9 x^{3 y})

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