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The derivative of the function/ cos(x^2+4*x+1)/(x+2)

Derivative of cos(x^2+4*x+1)/(x+2)

The solution

You have entered [src]

   / 2          \
cos\x  + 4*x + 1/
-----------------
      x + 2

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

cos(x^2 + 4*x + 1)/(x + 2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos{\left(\left(x^{2} + 4 x\right) + 1 \right)}$ and $g{\left(x \right)} = x + 2$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \left(x^{2} + 4 x\right) + 1$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} + 4 x\right) + 1\right)$ :
  1. Differentiate $\left(x^{2} + 4 x\right) + 1$ term by term:
    1. Differentiate $x^{2} + 4 x$ term by term:
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $4$
      The result is: $2 x + 4$
    2. The derivative of the constant $1$ is zero.
    The result is: $2 x + 4$
  The result of the chain rule is:
  
  $- \left(2 x + 4\right) \sin{\left(\left(x^{2} + 4 x\right) + 1 \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
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: $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                             /     2      \        /     2      \     /         2    /     2      \      /     2      \\                               \
  |       /     2      \   6*sin\1 + x  + 4*x/   3*cos\1 + x  + 4*x/   3*\2*(2 + x) *cos\1 + x  + 4*x/ + sin\1 + x  + 4*x//            2    /     2      \|
2*|- 6*cos\1 + x  + 4*x/ - ------------------- - ------------------- + ---------------------------------------------------- + 4*(2 + x) *sin\1 + x  + 4*x/|
  |                                     2                     4                                     2                                                     |
  \                              (2 + x)               (2 + x)                               (2 + x)                                                      /

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

Simplify

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Derivative of cos(x^2+4*x+1)/(x+2)

The graph:

Piecewise:

The solution