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Derivative of:
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Identical expressions
sqrt(two ^ two +(zero .04x^ two))
square root of (2 squared plus (0.04x squared ))
square root of (two to the power of two plus (zero .04x to the power of two))
√(2^2+(0.04x^2))
sqrt(22+(0.04x2))
sqrt22+0.04x2
sqrt(2²+(0.04x²))
sqrt(2 to the power of 2+(0.04x to the power of 2))
sqrt2^2+0.04x^2
Similar expressions
sqrt(2^2-(0.04x^2))

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

Derivative of sqrt(2^2+(0.04x^2))

The solution

You have entered [src]

     _________
    /       2 
   /   2   x  
  /   2  + -- 
\/         25

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

  /     _________\
  |    /       2 |
d |   /   2   x  |
--|  /   2  + -- |
dx\\/         25 /

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

Transform

Detail solution

Let $u = \frac{x^{2}}{25} + 2^{2}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\frac{x^{2}}{25} + 2^{2}\right)$ :
1. Differentiate $\frac{x^{2}}{25} + 2^{2}$ term by term:
  1. The derivative of the constant $2^{2}$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $\frac{2 x}{25}$
  The result is: $\frac{2 x}{25}$
The result of the chain rule is:

$\frac{x}{25 \sqrt{\frac{x^{2}}{25} + 2^{2}}}$
Now simplify:

$\frac{x}{5 \sqrt{x^{2} + 100}}$

The answer is:

$\frac{x}{5 \sqrt{x^{2} + 100}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        x        
-----------------
        _________
       /       2 
      /   2   x  
25*  /   2  + -- 
   \/         25

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

Simplify

The second derivative [src]

           2     
          x      
   25 - ------   
             2   
            x    
        4 + --   
            25   
-----------------
         ________
        /      2 
       /      x  
625*  /   4 + -- 
    \/        25

$$\frac{- \frac{x^{2}}{\frac{x^{2}}{25} + 4} + 25}{625 \sqrt{\frac{x^{2}}{25} + 4}}$$

Simplify

The third derivative [src]

    /         2  \
    |        x   |
3*x*|-25 + ------|
    |           2|
    |          x |
    |      4 + --|
    \          25/
------------------
              3/2 
      /     2\    
      |    x |    
15625*|4 + --|    
      \    25/

$$\frac{3 x \left(\frac{x^{2}}{\frac{x^{2}}{25} + 4} - 25\right)}{15625 \left(\frac{x^{2}}{25} + 4\right)^{\frac{3}{2}}}$$

Simplify

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Derivative of sqrt(2^2+(0.04x^2))

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The solution