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cosx/((x^ two)+ four)
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cosx/((x2)+4)
cosx/x2+4
cosx/((x²)+4)
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cosx/x^2+4
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Similar expressions
cosx/((x^2)-4)

The derivative of the function/ cosx/((x^2)+4)

Derivative of cosx/((x^2)+4)

The solution

You have entered [src]

cos(x)
------
 2    
x  + 4

$$\frac{\cos{\left(x \right)}}{x^{2} + 4}$$

cos(x)/(x^2 + 4)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = x^{2} + 4$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
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: $2 x$
Now plug in to the quotient rule:

$\frac{- 2 x \cos{\left(x \right)} - \left(x^{2} + 4\right) \sin{\left(x \right)}}{\left(x^{2} + 4\right)^{2}}$
Now simplify:

$- \frac{2 x \cos{\left(x \right)} + \left(x^{2} + 4\right) \sin{\left(x \right)}}{\left(x^{2} + 4\right)^{2}}$

The answer is:

$- \frac{2 x \cos{\left(x \right)} + \left(x^{2} + 4\right) \sin{\left(x \right)}}{\left(x^{2} + 4\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  sin(x)   2*x*cos(x)
- ------ - ----------
   2               2 
  x  + 4   / 2    \  
           \x  + 4/

$$- \frac{2 x \cos{\left(x \right)}}{\left(x^{2} + 4\right)^{2}} - \frac{\sin{\left(x \right)}}{x^{2} + 4}$$

Simplify

The second derivative [src]

            /         2 \                    
            |      4*x  |                    
          2*|-1 + ------|*cos(x)             
            |          2|                    
            \     4 + x /          4*x*sin(x)
-cos(x) + ---------------------- + ----------
                       2                  2  
                  4 + x              4 + x   
---------------------------------------------
                         2                   
                    4 + x

$$\frac{\frac{4 x \sin{\left(x \right)}}{x^{2} + 4} - \cos{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} + 4} - 1\right) \cos{\left(x \right)}}{x^{2} + 4}}{x^{2} + 4}$$

Simplify

The third derivative [src]

    /         2 \                            /         2 \                
    |      4*x  |                            |      2*x  |                
  6*|-1 + ------|*sin(x)                24*x*|-1 + ------|*cos(x)         
    |          2|                            |          2|                
    \     4 + x /          6*x*cos(x)        \     4 + x /                
- ---------------------- + ---------- - ------------------------- + sin(x)
               2                  2                     2                 
          4 + x              4 + x              /     2\                  
                                                \4 + x /                  
--------------------------------------------------------------------------
                                       2                                  
                                  4 + x

$$\frac{\frac{6 x \cos{\left(x \right)}}{x^{2} + 4} - \frac{24 x \left(\frac{2 x^{2}}{x^{2} + 4} - 1\right) \cos{\left(x \right)}}{\left(x^{2} + 4\right)^{2}} + \sin{\left(x \right)} - \frac{6 \left(\frac{4 x^{2}}{x^{2} + 4} - 1\right) \sin{\left(x \right)}}{x^{2} + 4}}{x^{2} + 4}$$

Simplify

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Derivative of cosx/((x^2)+4)

The graph:

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The solution