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Derivative of:
Derivative of x^4+4*x^3-8*x^2-5
Derivative of x^3/(2*x+4)
Derivative of x^(3/2)-3*x+1
Derivative of x^2*(x-2)
Identical expressions
one /(two sinx^2)
1 divide by (2 sinus of x squared )
one divide by (two sinus of x squared )
1/(2sinx2)
1/2sinx2
1/(2sinx²)
1/(2sinx to the power of 2)
1/2sinx^2
1 divide by (2sinx^2)

The derivative of the function/ 1/(2sinx^2)

Derivative of 1/(2sinx^2)

The solution

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

$$1 \cdot \frac{1}{2 \sin^{2}{\left(x \right)}}$$

d /      1    \
--|1*---------|
dx|       2   |
  \  2*sin (x)/

$$\frac{d}{d x} 1 \cdot \frac{1}{2 \sin^{2}{\left(x \right)}}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 1$ and $g{\left(x \right)} = 2 \sin^{2}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of the constant $1$ is zero.
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  So, the result is: $4 \sin{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

Derivative of 1/(2sinx^2)

The graph:

Piecewise:

The solution