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cot(x)/(x^3-x)
The derivative of the function/ cot(x)/(x^3-x)

Derivative of cot(x)/(x^3-x)

Function f() - derivative -N order at the point
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You have entered [src] 
cot(x)
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 3    
x  - x
cot⁡(x)x3−x\frac{\cot{\left(x \right)}}{x^{3} - x} 
d /cot(x)\
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dx| 3    |
  \x  - x/
ddxcot⁡(x)x3−x\frac{d}{d x} \frac{\cot{\left(x \right)}}{x^{3} - x}
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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)=cot⁡(x)f{\left(x \right)} = \cot{\left(x \right)} and g(x)=x3−xg{\left(x \right)} = x^{3} - x.

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

    1. There are multiple ways to do this derivative.

      Method #1

      1. Rewrite the function to be differentiated:

        cot⁡(x)=1tan⁡(x)\cot{\left(x \right)} = \frac{1}{\tan{\left(x \right)}}

      2. Let u=tan⁡(x)u = \tan{\left(x \right)}.

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

      4. Then, apply the chain rule. Multiply by ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

        1. Rewrite the function to be differentiated:

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

        2. 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)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

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

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

          1. The derivative of cosine is negative sine:

            ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

          Now plug in to the quotient rule:

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

        The result of the chain rule is:

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

      Method #2

      1. Rewrite the function to be differentiated:

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

      2. 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)=cos⁡(x)f{\left(x \right)} = \cos{\left(x \right)} 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 cosine is negative sine:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        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:

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

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

    1. Differentiate x3−xx^{3} - x term by term:

      1. Apply the power rule: x3x^{3} goes to 3x23 x^{2}

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: −1-1

      The result is: 3x2−13 x^{2} - 1

    Now plug in to the quotient rule:

    −(3x2−1)cot⁡(x)−(x3−x)(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)(x3−x)2\frac{- \left(3 x^{2} - 1\right) \cot{\left(x \right)} - \frac{\left(x^{3} - x\right) \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}}{\left(x^{3} - x\right)^{2}}

  2. Now simplify:

    −x(x2−1)+(1−3x2)sin⁡(2x)2x2(x2−1)2cos⁡2(x)tan⁡2(x)\frac{- x \left(x^{2} - 1\right) + \frac{\left(1 - 3 x^{2}\right) \sin{\left(2 x \right)}}{2}}{x^{2} \left(x^{2} - 1\right)^{2} \cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

The answer is:

−x(x2−1)+(1−3x2)sin⁡(2x)2x2(x2−1)2cos⁡2(x)tan⁡2(x)\frac{- x \left(x^{2} - 1\right) + \frac{\left(1 - 3 x^{2}\right) \sin{\left(2 x \right)}}{2}}{x^{2} \left(x^{2} - 1\right)^{2} \cos^{2}{\left(x \right)} \tan^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-50005000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        2      /       2\       
-1 - cot (x)   \1 - 3*x /*cot(x)
------------ + -----------------
    3                      2    
   x  - x          / 3    \     
                   \x  - x/
(−3x2+1)cot⁡(x)(x3−x)2+−cot⁡2(x)−1x3−x\frac{\left(- 3 x^{2} + 1\right) \cot{\left(x \right)}}{\left(x^{3} - x\right)^{2}} + \frac{- \cot^{2}{\left(x \right)} - 1}{x^{3} - x}
Simplify
The second derivative [src] 
  /                       /               2\                                   \
  |                       |    /        2\ |                                   |
  |                       |    \-1 + 3*x / |                                   |
  |                       |3 - ------------|*cot(x)                            |
  |                       |     2 /      2\|          /       2   \ /        2\|
  |/       2   \          \    x *\-1 + x //          \1 + cot (x)/*\-1 + 3*x /|
2*|\1 + cot (x)/*cot(x) - ------------------------- + -------------------------|
  |                                      2                     /      2\       |
  \                                -1 + x                    x*\-1 + x /       /
--------------------------------------------------------------------------------
                                    /      2\                                   
                                  x*\-1 + x /
2((cot⁡2(x)+1)cot⁡(x)−(3−(3x2−1)2x2(x2−1))cot⁡(x)x2−1+(3x2−1)(cot⁡2(x)+1)x(x2−1))x(x2−1)\frac{2 \left(\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)} - \frac{\left(3 - \frac{\left(3 x^{2} - 1\right)^{2}}{x^{2} \left(x^{2} - 1\right)}\right) \cot{\left(x \right)}}{x^{2} - 1} + \frac{\left(3 x^{2} - 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{x \left(x^{2} - 1\right)}\right)}{x \left(x^{2} - 1\right)}
Simplify
The third derivative [src] 
  /                                                                         /                                3\                                            \
  |                                                  /               2\     |      /        2\    /        2\ |                                            |
  |                                                  |    /        2\ |     |    6*\-1 + 3*x /    \-1 + 3*x / |                                            |
  |                                    /       2   \ |    \-1 + 3*x / |   3*|1 - ------------- + -------------|*cot(x)                                     |
  |                                  3*\1 + cot (x)/*|3 - ------------|     |             2                  2|                                            |
  |                                                  |     2 /      2\|     |       -1 + x        2 /      2\ |            /       2   \ /        2\       |
  |  /       2   \ /         2   \                   \    x *\-1 + x //     \                    x *\-1 + x / /          3*\1 + cot (x)/*\-1 + 3*x /*cot(x)|
2*|- \1 + cot (x)/*\1 + 3*cot (x)/ + ---------------------------------- - -------------------------------------------- - ----------------------------------|
  |                                                     2                                   /      2\                                 /      2\            |
  \                                               -1 + x                                  x*\-1 + x /                               x*\-1 + x /            /
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                                                                          /      2\                                                                         
                                                                        x*\-1 + x /
2⋅(−(cot⁡2(x)+1)(3cot⁡2(x)+1)+3⋅(3−(3x2−1)2x2(x2−1))(cot⁡2(x)+1)x2−1−3⋅(3x2−1)(cot⁡2(x)+1)cot⁡(x)x(x2−1)−3⋅(1−6⋅(3x2−1)x2−1+(3x2−1)3x2(x2−1)2)cot⁡(x)x(x2−1))x(x2−1)\frac{2 \cdot \left(- \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right) + \frac{3 \cdot \left(3 - \frac{\left(3 x^{2} - 1\right)^{2}}{x^{2} \left(x^{2} - 1\right)}\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{2} - 1} - \frac{3 \cdot \left(3 x^{2} - 1\right) \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x \left(x^{2} - 1\right)} - \frac{3 \cdot \left(1 - \frac{6 \cdot \left(3 x^{2} - 1\right)}{x^{2} - 1} + \frac{\left(3 x^{2} - 1\right)^{3}}{x^{2} \left(x^{2} - 1\right)^{2}}\right) \cot{\left(x \right)}}{x \left(x^{2} - 1\right)}\right)}{x \left(x^{2} - 1\right)}
Simplify
The graph
Derivative of cot(x)/(x^3-x)
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