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

The answer to (\partial)/(\partial x)(((x-y))/(ln(x/y))) is (xln(x/y)-x+y)/(xln^2(x/y))

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

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

\frac{x ln ( \frac{x}{y} ) - x + y}{x ln ^{2} ( \frac{x}{y} )}

Solution steps

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

Treat

y

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial x} ( x - y ) ln ( \frac{x}{y} ) - \frac{\partial}{\partial x} ( ln ( \frac{x}{y} ) ) ( x - y )}{( ln ( \frac{x}{y} ) ) ^{2}}

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

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

= \frac{1 \cdot ln ( \frac{x}{y} ) - \frac{1}{x} ( x - y )}{( ln ( \frac{x}{y} ) ) ^{2}}

Simplify

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

= \frac{x ln ( \frac{x}{y} ) - x + y}{x ln ^{2} ( \frac{x}{y} )}

2 \frac{d ^{2} y}{d x ^{2}} - 20 \frac{d y}{d x} + 48 y = 3 sin (5 x)

\frac{s}{s ^{2} + 2 s + 2}

\int (\frac{y}{2} - tan (x)) y d x

y^{^{''}} - 7 y + 10 y = 0

\int \frac{5}{1 + 5 x} d x

What is (\partial)/(\partial x)(((x-y))/(ln(x/y))) ?
The answer to (\partial)/(\partial x)(((x-y))/(ln(x/y))) is (xln(x/y)-x+y)/(xln^2(x/y))