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

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

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

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

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

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

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

Simplify

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

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

\int_{0}^{1} x + 2 d x

y^{^{''}} - y = 0, y (0) = 3

2 e^{x} + e^{2} x

3 + e^{- 0.1 t} cos (4 t)

\int 7 x cos (6 x) d x

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