What is the (x^{ln(x)})^' ?

The (x^{ln(x)})^' is (2e^{ln^2(x)}ln(x))/x

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

The first (x^{ln(x)})^' is (2e^{ln^2(x)}ln(x))/x

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(x^{l n (x)})^{^{'}}

\frac{2 e ^{l n^{2} (x)} ln ( x )}{x}

Solution steps

(x^{l n (x)})^{'}

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

(ln (x) ln (x))^{'} = \frac{2 ln ( x )}{x}

= e^{l n (x) l n (x)} \frac{2 ln ( x )}{x}

Simplify

e^{l n (x) l n (x)} \frac{2 ln ( x )}{x} : \frac{2 e ^{l n^{2} (x)} ln ( x )}{x}

= \frac{2 e ^{l n^{2} (x)} ln ( x )}{x}

What is the (x^{ln(x)})^' ?
The (x^{ln(x)})^' is (2e^{ln^2(x)}ln(x))/x
What is the first (x^{ln(x)})^' ?
The first (x^{ln(x)})^' is (2e^{ln^2(x)}ln(x))/x