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Similar expressions
cosx/(x-2)^2

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

Derivative of cosx/(x+2)^2

The solution

You have entered [src]

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

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

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

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 $g{\left(x \right)} = \left(x + 2\right)^{2}$ .

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. Let $u = x + 2$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 2\right)$ :
  1. Differentiate $x + 2$ term by term:
    1. The derivative of the constant $2$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $2 x + 4$
Now plug in to the quotient rule:

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

$- \frac{\left(x + 2\right) \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{\left(x + 2\right)^{3}}$

The answer is:

$- \frac{\left(x + 2\right) \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{\left(x + 2\right)^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

          4*sin(x)   6*cos(x)
-cos(x) + -------- + --------
           2 + x            2
                     (2 + x) 
-----------------------------
                  2          
           (2 + x)

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

Simplify

The third derivative [src]

  24*cos(x)   18*sin(x)   6*cos(x)         
- --------- - --------- + -------- + sin(x)
          3           2    2 + x           
   (2 + x)     (2 + x)                     
-------------------------------------------
                         2                 
                  (2 + x)

$$\frac{\sin{\left(x \right)} + \frac{6 \cos{\left(x \right)}}{x + 2} - \frac{18 \sin{\left(x \right)}}{\left(x + 2\right)^{2}} - \frac{24 \cos{\left(x \right)}}{\left(x + 2\right)^{3}}}{\left(x + 2\right)^{2}}$$

Simplify

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Derivative of cosx/(x+2)^2

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The solution