What is the derivative of x^{x/3} ?

The derivative of x^{x/3} is (x^{x/3}(ln(x)+1))/3

What is the first derivative of x^{x/3} ?

The first derivative of x^{x/3} is (x^{x/3}(ln(x)+1))/3

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derivative of

x^{\frac{x}{3}}

\frac{x ^{\frac{x}{3}} ( ln ( x ) + 1 )}{3}

Solution steps

\frac{d}{d x} (x^{\frac{x}{3}})

Apply exponent rule:

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

x^{\frac{x}{3}} = e^{\frac{x}{3} l n (x)}

= \frac{d}{d x} (e^{\frac{x}{3} l n (x)})

Apply the chain rule:

e^{\frac{x}{3} l n (x)} \frac{d}{d x} (\frac{x}{3} ln (x))

= e^{\frac{x}{3} l n (x)} \frac{d}{d x} (\frac{x}{3} ln (x))

\frac{d}{d x} (\frac{x}{3} ln (x)) = \frac{ln ( x ) + 1}{3}

= e^{\frac{x}{3} l n (x)} \frac{ln ( x ) + 1}{3}

Simplify

e^{\frac{x}{3} l n (x)} \frac{ln ( x ) + 1}{3} : \frac{x ^{\frac{x}{3}} ( ln ( x ) + 1 )}{3}

= \frac{x ^{\frac{x}{3}} ( ln ( x ) + 1 )}{3}

What is the derivative of x^{x/3} ?
The derivative of x^{x/3} is (x^{x/3}(ln(x)+1))/3
What is the first derivative of x^{x/3} ?
The first derivative of x^{x/3} is (x^{x/3}(ln(x)+1))/3