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Derivative of:
Derivative of 4/x^3
Derivative of atan(sqrt(x))
Derivative of -3x
Derivative of 4x-5
Identical expressions
(x^ two - six)*sqrt((four +x^ two))/120x
(x squared minus 6) multiply by square root of ((4 plus x squared )) divide by 120x
(x to the power of two minus six) multiply by square root of ((four plus x to the power of two)) divide by 120x
(x^2-6)*√((4+x^2))/120x
(x2-6)*sqrt((4+x2))/120x
x2-6*sqrt4+x2/120x
(x²-6)*sqrt((4+x²))/120x
(x to the power of 2-6)*sqrt((4+x to the power of 2))/120x
(x^2-6)sqrt((4+x^2))/120x
(x2-6)sqrt((4+x2))/120x
x2-6sqrt4+x2/120x
x^2-6sqrt4+x^2/120x
(x^2-6)*sqrt((4+x^2)) divide by 120x
Similar expressions
(x^2+6)*sqrt((4+x^2))/120x
(x^2-6)*sqrt((4-x^2))/120x

The derivative of the function/ (x^2-6)*sqrt((4+x^2))/120x

Derivative of (x^2-6)*sqrt((4+x^2))/120x

The solution

You have entered [src]

            ________  
/ 2    \   /      2   
\x  - 6/*\/  4 + x  *x
----------------------
         120

$$\frac{x \sqrt{x^{2} + 4} \left(x^{2} - 6\right)}{120}$$

  /            ________  \
  |/ 2    \   /      2   |
d |\x  - 6/*\/  4 + x  *x|
--|----------------------|
dx\         120          /

$$\frac{d}{d x} \frac{x \sqrt{x^{2} + 4} \left(x^{2} - 6\right)}{120}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} h{\left(x \right)} = f{\left(x \right)} g{\left(x \right)} \frac{d}{d x} h{\left(x \right)} + f{\left(x \right)} h{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} h{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \sqrt{x^{2} + 4}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = x^{2} + 4$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 4\right)$ :
    1. Differentiate $x^{2} + 4$ term by term:
      1. The derivative of the constant $4$ 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:
    
    $\frac{x}{\sqrt{x^{2} + 4}}$
  $h{\left(x \right)} = x^{2} - 6$ ; to find $\frac{d}{d x} h{\left(x \right)}$ :
  1. Differentiate $x^{2} - 6$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $\left(-1\right) 6$ is zero.
    The result is: $2 x$
  The result is: $2 x^{2} \sqrt{x^{2} + 4} + \frac{x^{2} \left(x^{2} - 6\right)}{\sqrt{x^{2} + 4}} + \sqrt{x^{2} + 4} \left(x^{2} - 6\right)$
So, the result is: $\frac{x^{2} \sqrt{x^{2} + 4}}{60} + \frac{x^{2} \left(x^{2} - 6\right)}{120 \sqrt{x^{2} + 4}} + \frac{\sqrt{x^{2} + 4} \left(x^{2} - 6\right)}{120}$
Now simplify:

$\frac{x^{4} - 6}{30 \sqrt{x^{2} + 4}}$

The answer is:

$\frac{x^{4} - 6}{30 \sqrt{x^{2} + 4}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      ________      ________                           
 2   /      2      /      2  / 2    \      2 / 2    \  
x *\/  4 + x     \/  4 + x  *\x  - 6/     x *\x  - 6/  
-------------- + -------------------- + ---------------
      60                 120                   ________
                                              /      2 
                                        120*\/  4 + x

$$\frac{x^{2} \sqrt{x^{2} + 4}}{60} + \frac{x^{2} \left(x^{2} - 6\right)}{120 \sqrt{x^{2} + 4}} + \frac{\sqrt{x^{2} + 4} \left(x^{2} - 6\right)}{120}$$

Simplify

The second derivative [src]

  /                                            /        2  \          \
  |                                            |       x   | /      2\|
  |                                            |-1 + ------|*\-6 + x /|
  |     ________     /      2\          2      |          2|          |
  |    /      2    2*\-6 + x /       4*x       \     4 + x /          |
x*|6*\/  4 + x   + ----------- + ----------- - -----------------------|
  |                   ________      ________            ________      |
  |                  /      2      /      2            /      2       |
  \                \/  4 + x     \/  4 + x           \/  4 + x        /
-----------------------------------------------------------------------
                                  120

$$\frac{x \left(\frac{4 x^{2}}{\sqrt{x^{2} + 4}} - \frac{\left(x^{2} - 6\right) \left(\frac{x^{2}}{x^{2} + 4} - 1\right)}{\sqrt{x^{2} + 4}} + \frac{2 \left(x^{2} - 6\right)}{\sqrt{x^{2} + 4}} + 6 \sqrt{x^{2} + 4}\right)}{120}$$

Simplify

The third derivative [src]

                              /        2  \                  /        2  \      /        2  \          
                              |       x   | /      2\      2 |       x   |    2 |       x   | /      2\
                              |-1 + ------|*\-6 + x /   2*x *|-1 + ------|   x *|-1 + ------|*\-6 + x /
     ________          2      |          2|                  |          2|      |          2|          
    /      2        6*x       \     4 + x /                  \     4 + x /      \     4 + x /          
2*\/  4 + x   + ----------- - ----------------------- - ------------------ + --------------------------
                   ________            ________               ________                      3/2        
                  /      2            /      2               /      2               /     2\           
                \/  4 + x           \/  4 + x              \/  4 + x                \4 + x /           
-------------------------------------------------------------------------------------------------------
                                                   40

$$\frac{\frac{x^{2} \left(x^{2} - 6\right) \left(\frac{x^{2}}{x^{2} + 4} - 1\right)}{\left(x^{2} + 4\right)^{\frac{3}{2}}} - \frac{2 x^{2} \left(\frac{x^{2}}{x^{2} + 4} - 1\right)}{\sqrt{x^{2} + 4}} + \frac{6 x^{2}}{\sqrt{x^{2} + 4}} - \frac{\left(x^{2} - 6\right) \left(\frac{x^{2}}{x^{2} + 4} - 1\right)}{\sqrt{x^{2} + 4}} + 2 \sqrt{x^{2} + 4}}{40}$$

Simplify

The graph

Derivative of (x^2-6)*sqrt((4+x^2))/120x

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Identical expressions

Similar expressions

Derivative of (x^2-6)*sqrt((4+x^2))/120x

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Piecewise:

The solution