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1/sin^4x
The derivative of the function/ 1/sin^4x

Derivative of 1/sin^4x

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

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The solution

You have entered [src] 
     1   
1*-------
     4   
  sin (x)
1⋅1sin⁡4(x)1 \cdot \frac{1}{\sin^{4}{\left(x \right)}} 
d /     1   \
--|1*-------|
dx|     4   |
  \  sin (x)/
ddx1⋅1sin⁡4(x)\frac{d}{d x} 1 \cdot \frac{1}{\sin^{4}{\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⁡4(x)g{\left(x \right)} = \sin^{4}{\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. Let u=sin⁡(x)u = \sin{\left(x \right)}.

    2. Apply the power rule: u4u^{4} goes to 4u34 u^{3}

    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:

      4sin⁡3(x)cos⁡(x)4 \sin^{3}{\left(x \right)} \cos{\left(x \right)}

    Now plug in to the quotient rule:

    −4cos⁡(x)sin⁡5(x)- \frac{4 \cos{\left(x \right)}}{\sin^{5}{\left(x \right)}}

The answer is:

−4cos⁡(x)sin⁡5(x)- \frac{4 \cos{\left(x \right)}}{\sin^{5}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-100000000000100000000000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  -4*cos(x)   
--------------
          4   
sin(x)*sin (x)
−4cos⁡(x)sin⁡(x)sin⁡4(x)- \frac{4 \cos{\left(x \right)}}{\sin{\left(x \right)} \sin^{4}{\left(x \right)}}
Simplify
The second derivative [src] 
  /         2   \
  |    5*cos (x)|
4*|1 + ---------|
  |        2    |
  \     sin (x) /
-----------------
        4        
     sin (x)
4⋅(1+5cos⁡2(x)sin⁡2(x))sin⁡4(x)\frac{4 \cdot \left(1 + \frac{5 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right)}{\sin^{4}{\left(x \right)}}
Simplify
The third derivative [src] 
   /          2   \       
   |    15*cos (x)|       
-8*|7 + ----------|*cos(x)
   |        2     |       
   \     sin (x)  /       
--------------------------
            5             
         sin (x)
−8⋅(7+15cos⁡2(x)sin⁡2(x))cos⁡(x)sin⁡5(x)- \frac{8 \cdot \left(7 + \frac{15 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin^{5}{\left(x \right)}}
Simplify
The graph
Derivative of 1/sin^4x
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