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Derivative of:
Derivative of 2
Derivative of 2*x^3
Derivative of ln(x^3)
Derivative of 2^(1/x)
Limit of the function:
2^(1/x)
Graphing y =:
2^(1/x)
Identical expressions
two ^(one /x)
2 to the power of (1 divide by x)
two to the power of (one divide by x)
2(1/x)
21/x
2^1/x
2^(1 divide by x)

The derivative of the function/ 2^(1/x)

Derivative of 2^(1/x)

The solution

You have entered [src]

x ___
\/ 2

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

2^(1/x)

Detail solution

Let $u = \frac{1}{x}$ .
$\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{1}{x}$ :
1. Apply the power rule: $\frac{1}{x}$ goes to $- \frac{1}{x^{2}}$
The result of the chain rule is:

$- \frac{2^{\frac{1}{x}} \log{\left(2 \right)}}{x^{2}}$

The answer is:

$- \frac{2^{\frac{1}{x}} \log{\left(2 \right)}}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x ___        
-\/ 2 *log(2) 
--------------
       2      
      x

$$- \frac{2^{\frac{1}{x}} \log{\left(2 \right)}}{x^{2}}$$

Simplify

The second derivative [src]

x ___ /    log(2)\       
\/ 2 *|2 + ------|*log(2)
      \      x   /       
-------------------------
             3           
            x

$$\frac{2^{\frac{1}{x}} \left(2 + \frac{\log{\left(2 \right)}}{x}\right) \log{\left(2 \right)}}{x^{3}}$$

Simplify

The third derivative [src]

       /       2              \        
 x ___ |    log (2)   6*log(2)|        
-\/ 2 *|6 + ------- + --------|*log(2) 
       |        2        x    |        
       \       x              /        
---------------------------------------
                    4                  
                   x

$$- \frac{2^{\frac{1}{x}} \left(6 + \frac{6 \log{\left(2 \right)}}{x} + \frac{\log{\left(2 \right)}^{2}}{x^{2}}\right) \log{\left(2 \right)}}{x^{4}}$$

Simplify

The graph

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Identical expressions

Derivative of 2^(1/x)

The graph:

Piecewise:

The solution