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

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

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

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

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

Solution steps

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{\partial}{\partial x} (e^{x^{2} + y^{2} + z^{2}})

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

\frac{\partial}{\partial y} (ln (x - 5 y))

x^{2} + 4 x + 3, y = - 2, - 2 < x < 0

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