What is the derivative of ln^x(x) ?

The derivative of ln^x(x) is ln^x(x)(ln(ln(x))+1/(ln(x)))

What is the first derivative of ln^x(x) ?

The first derivative of ln^x(x) is ln^x(x)(ln(ln(x))+1/(ln(x)))

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

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

Solution steps

\frac{d}{d 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{d}{d x} (e^{x l n (l n (x))})

Apply the chain rule:

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

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

\frac{d}{d 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 )})

\frac{d}{d t} (\frac{1}{e ^{t}})

f (x) = \frac{1 + 3 x}{1 - 3 x}

\int_{0}^{1} [x - x^{k}] d x

\frac{d y}{d x} - \frac{y}{2} = \frac{5}{2}

\frac{x}{1 - x}, 0

What is the derivative of ln^x(x) ?
The derivative of ln^x(x) is ln^x(x)(ln(ln(x))+1/(ln(x)))
What is the first derivative of ln^x(x) ?
The first derivative of ln^x(x) is ln^x(x)(ln(ln(x))+1/(ln(x)))