What is the derivative of x^{x/2} ?

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

What is the first derivative of x^{x/2} ?

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

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Popular >

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

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

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