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

Derivative of sin(x)^2/x^2

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

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The solution

You have entered [src] 
   2   
sin (x)
-------
    2  
   x
sin⁡2(x)x2\frac{\sin^{2}{\left(x \right)}}{x^{2}} 
sin(x)^2/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)=sin⁡2(x)f{\left(x \right)} = \sin^{2}{\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. Let u=sin⁡(x)u = \sin{\left(x \right)}.

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\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)}

      The result of the chain rule is:

      2sin⁡(x)cos⁡(x)2 \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:

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

  2. Now simplify:

    xsin⁡(2x)+cos⁡(2x)−1x3\frac{x \sin{\left(2 x \right)} + \cos{\left(2 x \right)} - 1}{x^{3}}

The answer is:

xsin⁡(2x)+cos⁡(2x)−1x3\frac{x \sin{\left(2 x \right)} + \cos{\left(2 x \right)} - 1}{x^{3}}

The graph
02468-8-6-4-2-10102-2
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       2                     
  2*sin (x)   2*cos(x)*sin(x)
- --------- + ---------------
       3              2      
      x              x
2sin⁡(x)cos⁡(x)x2−2sin⁡2(x)x3\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{2 \sin^{2}{\left(x \right)}}{x^{3}}
Simplify
The second derivative [src] 
  /                         2                     \
  |   2         2      3*sin (x)   4*cos(x)*sin(x)|
2*|cos (x) - sin (x) + --------- - ---------------|
  |                         2             x       |
  \                        x                      /
---------------------------------------------------
                          2                        
                         x
2(−sin⁡2(x)+cos⁡2(x)−4sin⁡(x)cos⁡(x)x+3sin⁡2(x)x2)x2\frac{2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{3 \sin^{2}{\left(x \right)}}{x^{2}}\right)}{x^{2}}
Simplify
The third derivative [src] 
  /       2                          /   2         2   \                  \
  |  6*sin (x)                     3*\sin (x) - cos (x)/   9*cos(x)*sin(x)|
4*|- --------- - 2*cos(x)*sin(x) + --------------------- + ---------------|
  |       3                                  x                     2      |
  \      x                                                        x       /
---------------------------------------------------------------------------
                                      2                                    
                                     x
4(−2sin⁡(x)cos⁡(x)+3(sin⁡2(x)−cos⁡2(x))x+9sin⁡(x)cos⁡(x)x2−6sin⁡2(x)x3)x2\frac{4 \left(- 2 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{x} + \frac{9 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{6 \sin^{2}{\left(x \right)}}{x^{3}}\right)}{x^{2}}
Simplify
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