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Integral of d{x}:
(1-cos(x))/x^2
Limit of the function:
(1-cos(x))/x^2
Identical expressions
(one -cos(x))/x^ two
(1 minus co sinus of e of (x)) divide by x squared
(one minus co sinus of e of (x)) divide by x to the power of two
(1-cos(x))/x2
1-cosx/x2
(1-cos(x))/x²
(1-cos(x))/x to the power of 2
1-cosx/x^2
(1-cos(x)) divide by x^2
Similar expressions
(1+cos(x))/x^2
(1-cosx)/x^2

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

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

The solution

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1 - cos(x)
----------
     2    
    x

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  6*(-1 + cos(x))   4*sin(x)         
- --------------- - -------- + cos(x)
          2            x             
         x                           
-------------------------------------
                   2                 
                  x

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

Simplify

The third derivative [src]

          6*cos(x)   18*sin(x)   24*(-1 + cos(x))
-sin(x) - -------- + --------- + ----------------
             x            2              3       
                         x              x        
-------------------------------------------------
                         2                       
                        x

$$\frac{- \sin{\left(x \right)} - \frac{6 \cos{\left(x \right)}}{x} + \frac{18 \sin{\left(x \right)}}{x^{2}} + \frac{24 \left(\cos{\left(x \right)} - 1\right)}{x^{3}}}{x^{2}}$$

Simplify

The graph

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Identical expressions

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

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Piecewise:

The solution