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(1+cos(8x))/(2x^2)

The derivative of the function/ (1-cos(8x))/(2x^2)

Derivative of (1-cos(8x))/(2x^2)

The solution

You have entered [src]

1 - cos(8*x)
------------
       2    
    2*x

$$\frac{1 - \cos{\left(8 x \right)}}{2 x^{2}}$$

(1 - cos(8*x))/((2*x^2))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 1 - \cos{\left(8 x \right)}$ and $g{\left(x \right)} = 2 x^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $1 - \cos{\left(8 x \right)}$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 8 x$ .
    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 $\frac{d}{d x} 8 x$ :
      1. 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: $8$
      The result of the chain rule is:
      
      $- 8 \sin{\left(8 x \right)}$
    So, the result is: $8 \sin{\left(8 x \right)}$
  The result is: $8 \sin{\left(8 x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. 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: $4 x$
Now plug in to the quotient rule:

$\frac{16 x^{2} \sin{\left(8 x \right)} - 4 x \left(1 - \cos{\left(8 x \right)}\right)}{4 x^{4}}$
Now simplify:

$\frac{4 x \sin{\left(8 x \right)} + \cos{\left(8 x \right)} - 1}{x^{3}}$

The answer is:

$\frac{4 x \sin{\left(8 x \right)} + \cos{\left(8 x \right)} - 1}{x^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1 - cos(8*x)      1           
- ------------ + 8*----*sin(8*x)
        3             2         
       x           2*x

$$8 \frac{1}{2 x^{2}} \sin{\left(8 x \right)} - \frac{1 - \cos{\left(8 x \right)}}{x^{3}}$$

Simplify

The second derivative [src]

              16*sin(8*x)   3*(-1 + cos(8*x))
32*cos(8*x) - ----------- - -----------------
                   x                 2       
                                    x        
---------------------------------------------
                       2                     
                      x

$$\frac{32 \cos{\left(8 x \right)} - \frac{16 \sin{\left(8 x \right)}}{x} - \frac{3 \left(\cos{\left(8 x \right)} - 1\right)}{x^{2}}}{x^{2}}$$

Simplify

The third derivative [src]

  /               48*cos(8*x)   3*(-1 + cos(8*x))   18*sin(8*x)\
4*|-64*sin(8*x) - ----------- + ----------------- + -----------|
  |                    x                 3                2    |
  \                                     x                x     /
----------------------------------------------------------------
                                2                               
                               x

$$\frac{4 \left(- 64 \sin{\left(8 x \right)} - \frac{48 \cos{\left(8 x \right)}}{x} + \frac{18 \sin{\left(8 x \right)}}{x^{2}} + \frac{3 \left(\cos{\left(8 x \right)} - 1\right)}{x^{3}}\right)}{x^{2}}$$

Simplify

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Derivative of (1-cos(8x))/(2x^2)

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The solution