Derivative of 1/(sinx)

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

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

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

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

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

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 sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

 /         2   \        
 |    6*cos (x)|        
-|5 + ---------|*cos(x) 
 |        2    |        
 \     sin (x) /        
------------------------
           2            
        sin (x)

$$- \frac{\left(5 + \frac{6 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$$

Simplify

The graph

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Derivative of 1/(sinx)

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The solution