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

The derivative of x^{ln(sqrt(x)}) is (e^{(ln^2(x))/2}ln(x))/(x)

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

The first derivative of x^{ln(sqrt(x)}) is (e^{(ln^2(x))/2}ln(x))/(x)

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

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

Solution steps

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (ln (x) ln (x)) = \frac{ln ( x )}{2 x} + \frac{ln ( x )}{x}

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

Simplify

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

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

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

\int \frac{4}{x ^{4}} d x

\frac{d}{d x} (16 + x^{2})

\frac{d}{d x} (a x cos (3 x))

n = 1 \sum \infty \frac{1}{2 ^{n} + 3 ^{n}}

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