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 )})

\int - 8 sin (x) d x

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(2 x + y e^{\frac{y}{x}}) d x - x e^{\frac{y}{x}} d y = 0, y (1) = 5

\frac{\partial}{\partial x} ((1 - y) + 10 (x - y^{2})^{2})

x y^{^{'}} + 3 y = x^{2} - 1

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)))