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

The answer to (\partial)/(\partial x)(1/(ln(y/x))) is 1/(xln^2(y/x))

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(\partial)/(\partial x)(1/(ln(y/x)))

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

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

Solution steps

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

Treat

y

as a constant

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{s ^{2}}{( s ^{2} + 1 ) ^{2}}

ln (\frac{( x + 6 ) ^{4} e ^{5 x^{3} - 3}}{x + 2 x ^{3}})

y = x^{2} + 4 x + 7, at x = 1

\frac{7}{2 7 x + 1}, at x = 9

ln (y^{2})

What is (\partial)/(\partial x)(1/(ln(y/x))) ?
The answer to (\partial)/(\partial x)(1/(ln(y/x))) is 1/(xln^2(y/x))