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

The derivative of y=(sqrt(x))^x is (sqrt(x))^x(ln(sqrt(x))+1/2)

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

The first derivative of y=(sqrt(x))^x is (sqrt(x))^x(ln(sqrt(x))+1/2)

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derivative of

y = (x)^{x}

(x)^{x} (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})

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

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

f (x) = 7 sin (x)

\frac{d}{d x} (\frac{- 2}{x})

\frac{d y}{d x} = 3 + 2 y

\int \frac{x}{x ^{2} - 16} d x

f (x) = x (x^{2} + 1)

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