What is the derivative of 2^{1/x} ?

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

What is the first derivative of 2^{1/x} ?

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

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derivative of

2^{\frac{1}{x}}

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

Solution steps

\frac{d}{d x} (2^{\frac{1}{x}})

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int \frac{3 x ^{2}}{( 25 + x ^{2} ) ^{2}} d x

\int sin^{10} (x) cos^{3} (x) d x

\int sin^{3} (2 x) cos^{2} (2 x) d x

x \to 1 lim (\frac{( sin ( x ) )}{x})

y^{^{'}} sin (x) = y ln (y)

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