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Derivative of:
Derivative of cos(x/3)
Derivative of √x
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Derivative of y=3x-2
Identical expressions
e^(x^ four / sixteen)/x
e to the power of (x to the power of 4 divide by 16) divide by x
e to the power of (x to the power of four divide by sixteen) divide by x
e(x4/16)/x
ex4/16/x
e^(x⁴/16)/x
e^x^4/16/x
e^(x^4 divide by 16) divide by x

The derivative of the function/ e^(x^4/16)/x

Derivative of e^(x^4/16)/x

The solution

You have entered [src]

  4
 x 
 --
 16
E  
---
 x

$$\frac{e^{\frac{x^{4}}{16}}}{x}$$

E^(x^4/16)/x

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = e^{\frac{x^{4}}{16}}$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \frac{x^{4}}{16}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x^{4}}{16}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{4}$ goes to $4 x^{3}$
    So, the result is: $\frac{x^{3}}{4}$
  The result of the chain rule is:
  
  $\frac{x^{3} e^{\frac{x^{4}}{16}}}{4}$
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:

$\frac{\frac{x^{4} e^{\frac{x^{4}}{16}}}{4} - e^{\frac{x^{4}}{16}}}{x^{2}}$
Now simplify:

$\frac{\left(x^{4} - 4\right) e^{\frac{x^{4}}{16}}}{4 x^{2}}$

The answer is:

$\frac{\left(x^{4} - 4\right) e^{\frac{x^{4}}{16}}}{4 x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    4        4
   x        x 
   --       --
   16    2  16
  e     x *e  
- --- + ------
    2     4   
   x

$$\frac{x^{2} e^{\frac{x^{4}}{16}}}{4} - \frac{e^{\frac{x^{4}}{16}}}{x^{2}}$$

Simplify

The second derivative [src]

                         4
                        x 
/           /      4\\  --
|2    x   x*\12 + x /|  16
|-- - - + -----------|*e  
| 3   2        16    |    
\x                   /

$$\left(\frac{x \left(x^{4} + 12\right)}{16} - \frac{x}{2} + \frac{2}{x^{3}}\right) e^{\frac{x^{4}}{16}}$$

Simplify

The third derivative [src]

                       4
                      x 
/          8      4\  --
|3   6    x    3*x |  16
|- - -- + -- + ----|*e  
|4    4   64    8  |    
\    x             /

$$\left(\frac{x^{8}}{64} + \frac{3 x^{4}}{8} + \frac{3}{4} - \frac{6}{x^{4}}\right) e^{\frac{x^{4}}{16}}$$

Simplify

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Identical expressions

Derivative of e^(x^4/16)/x

The graph:

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The solution