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

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

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

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

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derivative of

y = (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 )})

\int \frac{6 x ^{2} - 3 x}{x ^{2}} d x

y^{^{'}} = 2 t

f (x) = x + 11

\int - 4 e^{2 x} d x

\frac{d y}{d x} = e^{(y - x)}

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