What is (\partial)/(\partial x)((sqrt(x))^x) ?

The answer to (\partial)/(\partial x)((sqrt(x))^x) is x^{x/2}(1/2 ln(x)+1/2)

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(\partial)/(\partial x)((sqrt(x))^x)

\frac{\partial}{\partial x} ((x)^{x})

x^{\frac{x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

Solution steps

\frac{\partial}{\partial x} ((x)^{x})

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (x ln (x)) = ln (x) + \frac{1}{2}

= e^{x l n (x)} (ln (x) + \frac{1}{2})

Simplify

e^{x l n (x)} (ln (x) + \frac{1}{2}) : x^{\frac{x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

= x^{\frac{x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

3 e^{- x}

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What is (\partial)/(\partial x)((sqrt(x))^x) ?
The answer to (\partial)/(\partial x)((sqrt(x))^x) is x^{x/2}(1/2 ln(x)+1/2)