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

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

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(\partial)/(\partial x)((ln(x))^x)

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

ln^{x} (x) (ln (ln (x)) + \frac{1}{ln ( x )})

Solution steps

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

Apply exponent rule:

a^{b} = e^{b l n (a)}

(ln (x))^{x} = e^{x l n (l n (x))}

= \frac{\partial}{\partial x} (e^{x l n (l n (x))})

Apply the chain rule:

e^{x l n (l n (x))} \frac{\partial}{\partial x} (x ln (ln (x)))

= e^{x l n (l n (x))} \frac{\partial}{\partial x} (x ln (ln (x)))

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

= e^{x l n (l n (x))} (ln (ln (x)) + \frac{1}{ln ( x )})

e^{x l n (l n (x))} = ln^{x} (x)

= ln^{x} (x) (ln (ln (x)) + \frac{1}{ln ( x )})

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