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sinx^2/x^2

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

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

The solution

You have entered [src]

   2   
sin (x)
-------
    2  
   x

$$\frac{\sin^{2}{\left(x \right)}}{x^{2}}$$

sin(x)^2/x^2

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $2 \sin{\left(x \right)} \cos{\left(x \right)}$
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:

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

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

The answer is:

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

The graph

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

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

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

$$\frac{2 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} - \frac{4 \sin{\left(x \right)} \cos{\left(x \right)}}{x} + \frac{3 \sin^{2}{\left(x \right)}}{x^{2}}\right)}{x^{2}}$$

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

$$\frac{4 \left(- 2 \sin{\left(x \right)} \cos{\left(x \right)} + \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{x} + \frac{9 \sin{\left(x \right)} \cos{\left(x \right)}}{x^{2}} - \frac{6 \sin^{2}{\left(x \right)}}{x^{3}}\right)}{x^{2}}$$

Simplify

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

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The solution