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Derivative of:
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Integral of d{x}:
x/sin^2(x)
Identical expressions
x/sin^ two (x)
x divide by sinus of squared (x)
x divide by sinus of to the power of two (x)
x/sin2(x)
x/sin2x
x/sin²(x)
x/sin to the power of 2(x)
x/sin^2x
x divide by sin^2(x)

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

Derivative of x/sin^2(x)

The solution

You have entered [src]

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

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

x/sin(x)^2

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1      2*x*cos(x)
------- - ----------
   2          3     
sin (x)    sin (x)

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

Simplify

The second derivative [src]

  /  /         2   \           \
  |  |    3*cos (x)|   2*cos(x)|
2*|x*|1 + ---------| - --------|
  |  |        2    |    sin(x) |
  \  \     sin (x) /           /
--------------------------------
               2                
            sin (x)

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

Simplify

The third derivative [src]

  /                    /         2   \       \
  |                    |    3*cos (x)|       |
  |                4*x*|2 + ---------|*cos(x)|
  |         2          |        2    |       |
  |    9*cos (x)       \     sin (x) /       |
2*|3 + --------- - --------------------------|
  |        2                 sin(x)          |
  \     sin (x)                              /
----------------------------------------------
                      2                       
                   sin (x)

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

Simplify

Other calculators

How to use it?

Identical expressions

Derivative of x/sin^2(x)

The graph:

Piecewise:

The solution