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

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

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

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   1   
-------
   2   
sin (x)
1sin⁡2(x)\frac{1}{\sin^{2}{\left(x \right)}} 
1/(sin(x)^2)
Detail solution

  1. Let u=sin⁡2(x)u = \sin^{2}{\left(x \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 ddxsin⁡2(x)\frac{d}{d x} \sin^{2}{\left(x \right)}:

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

    2. Apply the power rule: u2u^{2} goes to 2u2 u

    3. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\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)}

      The result of the chain rule is:

      2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)}

    The result of the chain rule is:

    −2cos⁡(x)sin⁡3(x)- \frac{2 \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}

The answer is:

−2cos⁡(x)sin⁡3(x)- \frac{2 \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-50000005000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  -2*cos(x)   
--------------
          2   
sin(x)*sin (x)
−2cos⁡(x)sin⁡(x)sin⁡2(x)- \frac{2 \cos{\left(x \right)}}{\sin{\left(x \right)} \sin^{2}{\left(x \right)}}
Simplify
The second derivative [src] 
  /         2   \
  |    3*cos (x)|
2*|1 + ---------|
  |        2    |
  \     sin (x) /
-----------------
        2        
     sin (x)
2(1+3cos⁡2(x)sin⁡2(x))sin⁡2(x)\frac{2 \left(1 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{\sin^{2}{\left(x \right)}}
Simplify
The third derivative [src] 
   /         2   \       
   |    3*cos (x)|       
-8*|2 + ---------|*cos(x)
   |        2    |       
   \     sin (x) /       
-------------------------
            3            
         sin (x)
−8(2+3cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡3(x)- \frac{8 \left(2 + \frac{3 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}}
Simplify
The graph
Derivative of 1/(sin(x))^2
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