What is the derivative of x^{(1/5 ln(x)}) ?

The derivative of x^{(1/5 ln(x)}) is (2e^{(ln^2(x))/5}ln(x))/(5x)

What is the first derivative of x^{(1/5 ln(x)}) ?

The first derivative of x^{(1/5 ln(x)}) is (2e^{(ln^2(x))/5}ln(x))/(5x)

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\frac{d}{d x} (x^{(\frac{1}{5} l n (x))})

\frac{2 e ^{\frac{l n ^{2} ( x )}{5}} ln ( x )}{5 x}

Solution steps

\frac{d}{d x} (x^{(\frac{1}{5} l n (x))})

Apply exponent rule:

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

x^{\frac{1}{5} l n (x)} = e^{\frac{1}{5} l n (x) l n (x)}

= \frac{d}{d x} (e^{\frac{1}{5} l n (x) l n (x)})

Apply the chain rule:

e^{\frac{1}{5} l n (x) l n (x)} \frac{d}{d x} (\frac{1}{5} ln (x) ln (x))

= e^{\frac{1}{5} l n (x) l n (x)} \frac{d}{d x} (\frac{1}{5} ln (x) ln (x))

\frac{d}{d x} (\frac{1}{5} ln (x) ln (x)) = \frac{2 ln ( x )}{5 x}

= e^{\frac{1}{5} l n (x) l n (x)} \frac{2 ln ( x )}{5 x}

Simplify

e^{\frac{1}{5} l n (x) l n (x)} \frac{2 ln ( x )}{5 x} : \frac{2 e ^{\frac{l n ^{2} ( x )}{5}} ln ( x )}{5 x}

= \frac{2 e ^{\frac{l n ^{2} ( x )}{5}} ln ( x )}{5 x}

What is the derivative of x^{(1/5 ln(x)}) ?
The derivative of x^{(1/5 ln(x)}) is (2e^{(ln^2(x))/5}ln(x))/(5x)
What is the first derivative of x^{(1/5 ln(x)}) ?
The first derivative of x^{(1/5 ln(x)}) is (2e^{(ln^2(x))/5}ln(x))/(5x)