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sinx/x^2
The derivative of the function/ sinx/x^2

Derivative of sinx/x^2

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

You have entered [src] 
sin(x)
------
   2  
  x
sin⁡(x)x2\frac{\sin{\left(x \right)}}{x^{2}} 
d /sin(x)\
--|------|
dx|   2  |
  \  x   /
ddxsin⁡(x)x2\frac{d}{d x} \frac{\sin{\left(x \right)}}{x^{2}}
Transform
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)=sin⁡(x)f{\left(x \right)} = \sin{\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 sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\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:

    x2cos⁡(x)−2xsin⁡(x)x4\frac{x^{2} \cos{\left(x \right)} - 2 x \sin{\left(x \right)}}{x^{4}}

  2. Now simplify:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-200100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
cos(x)   2*sin(x)
------ - --------
   2         3   
  x         x
cos⁡(x)x2−2sin⁡(x)x3\frac{\cos{\left(x \right)}}{x^{2}} - \frac{2 \sin{\left(x \right)}}{x^{3}}
Simplify
The second derivative [src] 
          4*cos(x)   6*sin(x)
-sin(x) - -------- + --------
             x           2   
                        x    
-----------------------------
               2             
              x
−sin⁡(x)−4cos⁡(x)x+6sin⁡(x)x2x2\frac{- \sin{\left(x \right)} - \frac{4 \cos{\left(x \right)}}{x} + \frac{6 \sin{\left(x \right)}}{x^{2}}}{x^{2}}
Simplify
The third derivative [src] 
          24*sin(x)   6*sin(x)   18*cos(x)
-cos(x) - --------- + -------- + ---------
               3         x            2   
              x                      x    
------------------------------------------
                     2                    
                    x
−cos⁡(x)+6sin⁡(x)x+18cos⁡(x)x2−24sin⁡(x)x3x2\frac{- \cos{\left(x \right)} + \frac{6 \sin{\left(x \right)}}{x} + \frac{18 \cos{\left(x \right)}}{x^{2}} - \frac{24 \sin{\left(x \right)}}{x^{3}}}{x^{2}}
Simplify
The graph
Derivative of sinx/x^2
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