What is (\partial)/(\partial x)(2^{1/x}) ?

The answer to (\partial)/(\partial x)(2^{1/x}) is -(ln(2)*2^{1/x})/(x^2)

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(\partial)/(\partial x)(2^{1/x})

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

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

Solution steps

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

Apply the chain rule:

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

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

\frac{\partial}{\partial 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}}

(3, 5), (8, 7)

\int \frac{3 x ^{2} - 6 x ^{2} + x}{x ^{3}} d x

y = 5 x + 20, at x = 1

f (x) = 16 x + 3

What is (\partial)/(\partial x)(2^{1/x}) ?
The answer to (\partial)/(\partial x)(2^{1/x}) is -(ln(2)*2^{1/x})/(x^2)