What is (\partial)/(\partial x)((2pikt)/(ln(r/x))) ?

The answer to (\partial)/(\partial x)((2pikt)/(ln(r/x))) is (2pikt)/(xln^2(r/x))

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(\partial)/(\partial x)((2pikt)/(ln(r/x)))

\frac{\partial}{\partial x} (\frac{2 πk t}{ln ( \frac{r}{x} )})

\frac{2 πk t}{x ln ^{2} ( \frac{r}{x} )}

Solution steps

\frac{\partial}{\partial x} (\frac{2 πk t}{ln ( \frac{r}{x} )})

Treat

k, t, r

as constants

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 2 πk t \frac{\partial}{\partial x} (\frac{1}{ln ( \frac{r}{x} )})

Apply exponent rule:

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

= 2 πk t \frac{\partial}{\partial x} (ln^{- 1} (\frac{r}{x}))

Apply the chain rule:

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

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

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

= 2 πk t (- \frac{1}{( ln ( \frac{r}{x} ) ) ^{2}} (- \frac{1}{x}))

Simplify

2 πk t (- \frac{1}{ln ^{2} ( \frac{r}{x} )} (- \frac{1}{x})) : \frac{2 πk t}{x ln ^{2} ( \frac{r}{x} )}

= \frac{2 πk t}{x ln ^{2} ( \frac{r}{x} )}

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