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

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

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
1 - cos(x)
----------
     2    
    x
1−cos⁡(x)x2\frac{1 - \cos{\left(x \right)}}{x^{2}} 
(1 - cos(x))/x^2
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=1−cos⁡(x)f{\left(x \right)} = 1 - \cos{\left(x \right)} and g(x)=x2g{\left(x \right)} = x^{2}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate 1−cos⁡(x)1 - \cos{\left(x \right)} term by term:

      1. The derivative of the constant 11 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:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        So, the result is: sin⁡(x)\sin{\left(x \right)}

      The result is: sin⁡(x)\sin{\left(x \right)}

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Apply the power rule: x2x^{2} goes to 2x2 x

    Now plug in to the quotient rule:

    x2sin⁡(x)−2x(1−cos⁡(x))x4\frac{x^{2} \sin{\left(x \right)} - 2 x \left(1 - \cos{\left(x \right)}\right)}{x^{4}}

  2. Now simplify:

    xsin⁡(x)+2cos⁡(x)−2x3\frac{x \sin{\left(x \right)} + 2 \cos{\left(x \right)} - 2}{x^{3}}

The answer is:

xsin⁡(x)+2cos⁡(x)−2x3\frac{x \sin{\left(x \right)} + 2 \cos{\left(x \right)} - 2}{x^{3}}

The graph
02468-8-6-4-2-10101.0-0.5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
sin(x)   2*(1 - cos(x))
------ - --------------
   2            3      
  x            x
sin⁡(x)x2−2(1−cos⁡(x))x3\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
cos⁡(x)−4sin⁡(x)x−6(cos⁡(x)−1)x2x2\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
−sin⁡(x)−6cos⁡(x)x+18sin⁡(x)x2+24(cos⁡(x)−1)x3x2\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
Derivative of (1-cos(x))/x^2
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