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

Derivative of 2^(1/x)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
x ___
\/ 2
21x2^{\frac{1}{x}} 
2^(1/x)
Detail solution

  1. Let u=1xu = \frac{1}{x}.

  2. ddu2u=2ulog⁡(2)\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}

  3. Then, apply the chain rule. Multiply by ddx1x\frac{d}{d x} \frac{1}{x}:

    1. Apply the power rule: 1x\frac{1}{x} goes to −1x2- \frac{1}{x^{2}}

    The result of the chain rule is:

    −21xlog⁡(2)x2- \frac{2^{\frac{1}{x}} \log{\left(2 \right)}}{x^{2}}

The answer is:

−21xlog⁡(2)x2- \frac{2^{\frac{1}{x}} \log{\left(2 \right)}}{x^{2}}

The graph
02468-8-6-4-2-1010-10000050000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
 x ___        
-\/ 2 *log(2) 
--------------
       2      
      x
−21xlog⁡(2)x2- \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
21x(2+log⁡(2)x)log⁡(2)x3\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
−21x(6+6log⁡(2)x+log⁡(2)2x2)log⁡(2)x4- \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
Derivative of 2^(1/x)
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