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

Derivative of 1/(sinx)

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

The graph:

from to

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The solution

You have entered [src] 
    1   
1*------
  sin(x)
1⋅1sin⁡(x)1 \cdot \frac{1}{\sin{\left(x \right)}} 
d /    1   \
--|1*------|
dx\  sin(x)/
ddx1⋅1sin⁡(x)\frac{d}{d x} 1 \cdot \frac{1}{\sin{\left(x \right)}}
Transform
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=1f{\left(x \right)} = 1 and g(x)=sin⁡(x)g{\left(x \right)} = \sin{\left(x \right)}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of the constant 11 is zero.

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. The derivative of sine is cosine:

      ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

    Now plug in to the quotient rule:

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

The answer is:

−cos⁡(x)sin⁡2(x)- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \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(x) 
--------
   2    
sin (x)
−cos⁡(x)sin⁡2(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)
1+2cos⁡2(x)sin⁡2(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)
−(5+6cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡2(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
Derivative of 1/(sinx)
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