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The derivative of the function/ cot(x)/(x^4-16)

Derivative of cot(x)/(x^4-16)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
 cot(x)
-------
 4     
x  - 16
cot⁡(x)x4−16\frac{\cot{\left(x \right)}}{x^{4} - 16} 
cot(x)/(x^4 - 16)
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)=x4−16g{\left(x \right)} = x^{4} - 16.

    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 x4−16x^{4} - 16 term by term:

      1. The derivative of the constant −16-16 is zero.

      2. Apply the power rule: x4x^{4} goes to 4x34 x^{3}

      The result is: 4x34 x^{3}

    Now plug in to the quotient rule:

    −4x3cot⁡(x)−(x4−16)(sin⁡2(x)+cos⁡2(x))cos⁡2(x)tan⁡2(x)(x4−16)2\frac{- 4 x^{3} \cot{\left(x \right)} - \frac{\left(x^{4} - 16\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^{4} - 16\right)^{2}}

  2. Now simplify:

    2(−x4−2x3sin⁡(2x)+16)(1−cos⁡(2x))(x4−16)2\frac{2 \left(- x^{4} - 2 x^{3} \sin{\left(2 x \right)} + 16\right)}{\left(1 - \cos{\left(2 x \right)}\right) \left(x^{4} - 16\right)^{2}}

The answer is:

2(−x4−2x3sin⁡(2x)+16)(1−cos⁡(2x))(x4−16)2\frac{2 \left(- x^{4} - 2 x^{3} \sin{\left(2 x \right)} + 16\right)}{\left(1 - \cos{\left(2 x \right)}\right) \left(x^{4} - 16\right)^{2}}

The graph
02468-8-6-4-2-1010-200100
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
        2         3       
-1 - cot (x)   4*x *cot(x)
------------ - -----------
   4                     2
  x  - 16       / 4     \ 
                \x  - 16/
−4x3cot⁡(x)(x4−16)2+−cot⁡2(x)−1x4−16- \frac{4 x^{3} \cot{\left(x \right)}}{\left(x^{4} - 16\right)^{2}} + \frac{- \cot^{2}{\left(x \right)} - 1}{x^{4} - 16}
Simplify
The second derivative [src] 
  /                                                 /          4  \       \
  |                                               2 |       8*x   |       |
  |                                            2*x *|-3 + --------|*cot(x)|
  |                          3 /       2   \        |            4|       |
  |/       2   \          4*x *\1 + cot (x)/        \     -16 + x /       |
2*|\1 + cot (x)/*cot(x) + ------------------ + ---------------------------|
  |                                   4                         4         |
  \                            -16 + x                   -16 + x          /
---------------------------------------------------------------------------
                                         4                                 
                                  -16 + x
2(4x3(cot⁡2(x)+1)x4−16+2x2(8x4x4−16−3)cot⁡(x)x4−16+(cot⁡2(x)+1)cot⁡(x))x4−16\frac{2 \left(\frac{4 x^{3} \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{4} - 16} + \frac{2 x^{2} \left(\frac{8 x^{4}}{x^{4} - 16} - 3\right) \cot{\left(x \right)}}{x^{4} - 16} + \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}\right)}{x^{4} - 16}
Simplify
The third derivative [src] 
   /                                                                          /         4            8   \                                    \
   |                                                   /          4  \        |     12*x         16*x    |                                    |
   |                                   2 /       2   \ |       8*x   |   12*x*|1 - -------- + -----------|*cot(x)                             |
   |                                6*x *\1 + cot (x)/*|-3 + --------|        |           4             2|                                    |
   |                                                   |            4|        |    -16 + x    /       4\ |              3 /       2   \       |
   |/       2   \ /         2   \                      \     -16 + x /        \               \-16 + x / /          12*x *\1 + cot (x)/*cot(x)|
-2*|\1 + cot (x)/*\1 + 3*cot (x)/ + ---------------------------------- + ---------------------------------------- + --------------------------|
   |                                                    4                                       4                                   4         |
   \                                             -16 + x                                 -16 + x                             -16 + x          /
-----------------------------------------------------------------------------------------------------------------------------------------------
                                                                           4                                                                   
                                                                    -16 + x
−2(12x3(cot⁡2(x)+1)cot⁡(x)x4−16+6x2(8x4x4−16−3)(cot⁡2(x)+1)x4−16+12x(16x8(x4−16)2−12x4x4−16+1)cot⁡(x)x4−16+(cot⁡2(x)+1)(3cot⁡2(x)+1))x4−16- \frac{2 \left(\frac{12 x^{3} \left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{x^{4} - 16} + \frac{6 x^{2} \left(\frac{8 x^{4}}{x^{4} - 16} - 3\right) \left(\cot^{2}{\left(x \right)} + 1\right)}{x^{4} - 16} + \frac{12 x \left(\frac{16 x^{8}}{\left(x^{4} - 16\right)^{2}} - \frac{12 x^{4}}{x^{4} - 16} + 1\right) \cot{\left(x \right)}}{x^{4} - 16} + \left(\cot^{2}{\left(x \right)} + 1\right) \left(3 \cot^{2}{\left(x \right)} + 1\right)\right)}{x^{4} - 16}
Simplify
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