What is the derivative of sqrt(x)^x ?

The derivative of sqrt(x)^x is x^{x/2}(1/2 ln(x)+1/2)

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

The first derivative of sqrt(x)^x is x^{x/2}(1/2 ln(x)+1/2)

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d 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})

\int \frac{x + 2}{x - 4} d x

\frac{d}{d x} (cos (e^{2 x}))

9 x - 8 x

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

x \to 2 lim (cos (x - 2))

What is the derivative of sqrt(x)^x ?
The derivative of sqrt(x)^x is x^{x/2}(1/2 ln(x)+1/2)
What is the first derivative of sqrt(x)^x ?
The first derivative of sqrt(x)^x is x^{x/2}(1/2 ln(x)+1/2)