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Derivative of:
Derivative of 5
Derivative of 3
Derivative of x*sqrt(x)
Derivative of 1-x
Integral of d{x}:
1/(2*sin(x))
Identical expressions
one /(two *sin(x))
1 divide by (2 multiply by sinus of (x))
one divide by (two multiply by sinus of (x))
1/(2sin(x))
1/2sinx
1 divide by (2*sin(x))
Similar expressions
y=(x^1/2)(sinx)
y=-5√x+1/2sinx-1/4tgx
2^1/2(sinx-cosx)-sin^2x-x
-1/2sinx/2
1/(2sinx^2)
1/(2*sinx)

The derivative of the function/ 1/(2*sin(x))

Derivative of 1/(2*sin(x))

The solution

You have entered [src]

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

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

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

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

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

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. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  So, the result is: $2 \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

 /         2   \        
 |5   3*cos (x)|        
-|- + ---------|*cos(x) 
 |2       2    |        
 \     sin (x) /        
------------------------
           2            
        sin (x)

$$- \frac{\left(\frac{5}{2} + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

Simplify

The graph

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Derivative of 1/(2*sin(x))

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The solution