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

Derivative of cos(x)/x^2

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
cos(x)
------
   2  
  x
cos⁡(x)x2\frac{\cos{\left(x \right)}}{x^{2}} 
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)=cos⁡(x)f{\left(x \right)} = \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. The derivative of cosine is negative sine:

      ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \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)−2xcos⁡(x)x4\frac{- x^{2} \sin{\left(x \right)} - 2 x \cos{\left(x \right)}}{x^{4}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-50005000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  sin(x)   2*cos(x)
- ------ - --------
     2         3   
    x         x
−sin⁡(x)x2−2cos⁡(x)x3- \frac{\sin{\left(x \right)}}{x^{2}} - \frac{2 \cos{\left(x \right)}}{x^{3}}
Simplify
The second derivative [src] 
          4*sin(x)   6*cos(x)
-cos(x) + -------- + --------
             x           2   
                        x    
-----------------------------
               2             
              x
−cos⁡(x)+4sin⁡(x)x+6cos⁡(x)x2x2\frac{- \cos{\left(x \right)} + \frac{4 \sin{\left(x \right)}}{x} + \frac{6 \cos{\left(x \right)}}{x^{2}}}{x^{2}}
Simplify
The third derivative [src] 
  24*cos(x)   18*sin(x)   6*cos(x)         
- --------- - --------- + -------- + sin(x)
       3           2         x             
      x           x                        
-------------------------------------------
                      2                    
                     x
sin⁡(x)+6cos⁡(x)x−18sin⁡(x)x2−24cos⁡(x)x3x2\frac{\sin{\left(x \right)} + \frac{6 \cos{\left(x \right)}}{x} - \frac{18 \sin{\left(x \right)}}{x^{2}} - \frac{24 \cos{\left(x \right)}}{x^{3}}}{x^{2}}
Simplify
The graph
Derivative of cos(x)/x^2
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