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Derivative of:
Derivative of (-6)/x^4
Derivative of ln(x+5)^9
Derivative of 4*cos(x)-tan(x)
Derivative of 4^(1/x)
Graphing y =:
4^(1/x)
Limit of the function:
4^(1/x)
Identical expressions
four ^(one /x)
4 to the power of (1 divide by x)
four to the power of (one divide by x)
4(1/x)
41/x
4^1/x
4^(1 divide by x)

The derivative of the function/ 4^(1/x)

Derivative of 4^(1/x)

The solution

You have entered [src]

x ___
\/ 4

4^{1 \cdot \frac{1}{x}}

d /x ___\
--\\/ 4 /
dx

\frac{d}{d x} 4^{1 \cdot \frac{1}{x}}

Transform

Detail solution

Let $u = 1 \cdot \frac{1}{x}$ .
$\frac{d}{d u} 4^{u} = 4^{u} \log{\left(4 \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} 1 \cdot \frac{1}{x}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 1$ and $g{\left(x \right)} = x$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of the constant $1$ is zero.
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  Now plug in to the quotient rule:
  
  $- \frac{1}{x^{2}}$
The result of the chain rule is:

$- \frac{4^{\frac{1}{x}} \log{\left(4 \right)}}{x^{2}}$
Now simplify:

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

The answer is:

- \frac{2^{\frac{2}{x}} \log{\left(4 \right)}}{x^{2}}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 x ___        
-\/ 4 *log(4) 
--------------
       2      
      x

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

Simplify

The second derivative [src]

x ___ /    log(4)\       
\/ 4 *|2 + ------|*log(4)
      \      x   /       
-------------------------
             3           
            x

\frac{4^{\frac{1}{x}} \left(2 + \frac{\log{\left(4 \right)}}{x}\right) \log{\left(4 \right)}}{x^{3}}

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of 4^(1/x)

The graph:

Piecewise:

The solution