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1/sin(y)
The derivative of the function/ 1/sin(y)

Derivative of 1/sin(y)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1   
------
sin(y)
1sin⁡(y)\frac{1}{\sin{\left(y \right)}} 
1/sin(y)
Detail solution

  1. Let u=sin⁡(y)u = \sin{\left(y \right)}.

  2. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

  3. Then, apply the chain rule. Multiply by ddysin⁡(y)\frac{d}{d y} \sin{\left(y \right)}:

    1. The derivative of sine is cosine:

      ddysin⁡(y)=cos⁡(y)\frac{d}{d y} \sin{\left(y \right)} = \cos{\left(y \right)}

    The result of the chain rule is:

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

The answer is:

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

The graph
02468-8-6-4-2-1010-1000010000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
-cos(y) 
--------
   2    
sin (y)
−cos⁡(y)sin⁡2(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)
1+2cos⁡2(y)sin⁡2(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)
−(5+6cos⁡2(y)sin⁡2(y))cos⁡(y)sin⁡2(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
Derivative of 1/sin(y)
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