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

The derivative of (ln(x)^{ln(x)}) is x^{ln(ln(x))-1}(ln(ln(x))+1)

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

The first derivative of (ln(x)^{ln(x)}) is x^{ln(ln(x))-1}(ln(ln(x))+1)

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

x^{l n (l n (x)) - 1} (ln (ln (x)) + 1)

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

= x^{l n (l n (x)) - 1} (ln (ln (x)) + 1)

\int \frac{1}{12 x ^{5}} d x

x \to - \infty + lim (- 5 x^{3})

\frac{d}{d x} (\frac{x ^{6}}{4 - x ^{5}})

\frac{d}{d x} (3 x^{8})

\int e^{2 x} 1 + e^{2 x} d x

What is the derivative of (ln(x)^{ln(x)}) ?
The derivative of (ln(x)^{ln(x)}) is x^{ln(ln(x))-1}(ln(ln(x))+1)
What is the first derivative of (ln(x)^{ln(x)}) ?
The first derivative of (ln(x)^{ln(x)}) is x^{ln(ln(x))-1}(ln(ln(x))+1)