Derivative of 1/sin(y)

You have entered [src]

  1   
------
sin(y)

$$\frac{1}{\sin{\left(y \right)}}$$

1/sin(y)

Detail solution

Let $u = \sin{\left(y \right)}$ .
Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
Then, apply the chain rule. Multiply by $\frac{d}{d y} \sin{\left(y \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d y} \sin{\left(y \right)} = \cos{\left(y \right)}$
The result of the chain rule is:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-cos(y) 
--------
   2    
sin (y)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Derivative of 1/sin(y)

The graph:

Piecewise:

The solution