What is the derivative of a^{1/x} ?

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

What is the first derivative of a^{1/x} ?

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

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derivative of

a^{\frac{1}{x}}

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

Solution steps

\frac{d}{d x} (a^{\frac{1}{x}})

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

d er i v a t i v e 3 ln (2 x - 1)

d er i v a t i v e y = x^{2} ln (4 x)

d er i v a t i v e 2^{3} x

d er i v a t i v e y = x^{2} e^{2}

d er i v a t i v e lo g_{10} (x^{2} + a^{2})

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