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

Derivative of sinx^2/sin^2x

The solution

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

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

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

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. 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:
  
  $2 \sin{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

$0$

The answer is:

$0$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  2*cos(x)   2*cos(x)*sin(x)
- -------- + ---------------
   sin(x)           2       
                 sin (x)

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

Simplify

The second derivative [src]

  /       2         2         2   \
  |    cos (x)   sin (x) - cos (x)|
2*|1 - ------- - -----------------|
  |       2              2        |
  \    sin (x)        sin (x)     /

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

Simplify

The third derivative [src]

  /          2        /   2         2   \\       
  |     3*cos (x)   3*\sin (x) - cos (x)/|       
4*|-3 + --------- + ---------------------|*cos(x)
  |         2                 2          |       
  \      sin (x)           sin (x)       /       
-------------------------------------------------
                      sin(x)

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

Simplify

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Derivative of sinx^2/sin^2x

The graph:

Piecewise:

The solution