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zero . two x/(zero .04x^ two + one)^(one /2)
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Similar expressions
0.2x/(0.04x^2-1)^(1/2)

The derivative of the function/ 0.2x/(0.04x^2+1)^(1/2)

Derivative of 0.2x/(0.04x^2+1)^(1/2)

The solution

You have entered [src]

       x       
---------------
       ________
      /  2     
     /  x      
5*  /   -- + 1 
  \/    25

$$\frac{x}{5 \sqrt{\frac{x^{2}}{25} + 1}}$$

d /       x       \
--|---------------|
dx|       ________|
  |      /  2     |
  |     /  x      |
  |5*  /   -- + 1 |
  \  \/    25     /

$$\frac{d}{d x} \frac{x}{5 \sqrt{\frac{x^{2}}{25} + 1}}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 5 x$ and $g{\left(x \right)} = \sqrt{x^{2} + 25}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x$ goes to $1$
    So, the result is: $5$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = x^{2} + 25$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 25\right)$ :
    1. Differentiate $x^{2} + 25$ term by term:
      1. The derivative of the constant $25$ is zero.
      2. Apply the power rule: $x^{2}$ goes to $2 x$
      The result is: $2 x$
    The result of the chain rule is:
    
    $\frac{x}{\sqrt{x^{2} + 25}}$
  Now plug in to the quotient rule:
  
  $\frac{- \frac{5 x^{2}}{\sqrt{x^{2} + 25}} + 5 \sqrt{x^{2} + 25}}{x^{2} + 25}$
So, the result is: $\frac{- \frac{5 x^{2}}{\sqrt{x^{2} + 25}} + 5 \sqrt{x^{2} + 25}}{5 \left(x^{2} + 25\right)}$
Now simplify:

$\frac{25}{\left(x^{2} + 25\right)^{\frac{3}{2}}}$

The answer is:

$\frac{25}{\left(x^{2} + 25\right)^{\frac{3}{2}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                          2      
       1                 x       
--------------- - ---------------
       ________               3/2
      /  2            / 2    \   
     /  x             |x     |   
5*  /   -- + 1    125*|-- + 1|   
  \/    25            \25    /

$$- \frac{x^{2}}{125 \left(\frac{x^{2}}{25} + 1\right)^{\frac{3}{2}}} + \frac{1}{5 \sqrt{\frac{x^{2}}{25} + 1}}$$

Simplify

The second derivative [src]

  /          2 \
  |       3*x  |
x*|-3 + -------|
  |           2|
  \     25 + x /
----------------
            3/2 
    /     2\    
    |    x |    
125*|1 + --|    
    \    25/

$$\frac{x \left(\frac{3 x^{2}}{x^{2} + 25} - 3\right)}{125 \left(\frac{x^{2}}{25} + 1\right)^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

  /                   /          2 \\
  |                 2 |       5*x  ||
  |                x *|-3 + -------||
  |           2       |           2||
  |       75*x        \     25 + x /|
3*|-25 + ------- - -----------------|
  |            2              2     |
  |      25 + x              x      |
  |                      1 + --     |
  \                          25     /
-------------------------------------
                        3/2          
                /     2\             
                |    x |             
           3125*|1 + --|             
                \    25/

$$\frac{3 \cdot \left(- \frac{x^{2} \cdot \left(\frac{5 x^{2}}{x^{2} + 25} - 3\right)}{\frac{x^{2}}{25} + 1} + \frac{75 x^{2}}{x^{2} + 25} - 25\right)}{3125 \left(\frac{x^{2}}{25} + 1\right)^{\frac{3}{2}}}$$

Simplify

The graph

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Derivative of 0.2x/(0.04x^2+1)^(1/2)

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