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Derivative of ln(ctg^2)*x

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

The graph:

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Piecewise:

The solution

You have entered [src]
   /   2   \  
log\cot (x)/*x
$$x \log{\left(\cot^{2}{\left(x \right)} \right)}$$
log(cot(x)^2)*x
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Let .

    2. The derivative of is .

    3. Then, apply the chain rule. Multiply by :

      1. Let .

      2. Apply the power rule: goes to

      3. Then, apply the chain rule. Multiply by :

        1. There are multiple ways to do this derivative.

          Method #1

          1. Rewrite the function to be differentiated:

          2. Let .

          3. Apply the power rule: goes to

          4. Then, apply the chain rule. Multiply by :

            1. Rewrite the function to be differentiated:

            2. Apply the quotient rule, which is:

              and .

              To find :

              1. The derivative of sine is cosine:

              To find :

              1. The derivative of cosine is negative sine:

              Now plug in to the quotient rule:

            The result of the chain rule is:

          Method #2

          1. Rewrite the function to be differentiated:

          2. Apply the quotient rule, which is:

            and .

            To find :

            1. The derivative of cosine is negative sine:

            To find :

            1. The derivative of sine is cosine:

            Now plug in to the quotient rule:

        The result of the chain rule is:

      The result of the chain rule is:

    ; to find :

    1. Apply the power rule: goes to

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
  /          2   \               
x*\-2 - 2*cot (x)/      /   2   \
------------------ + log\cot (x)/
      cot(x)                     
$$\frac{x \left(- 2 \cot^{2}{\left(x \right)} - 2\right)}{\cot{\left(x \right)}} + \log{\left(\cot^{2}{\left(x \right)} \right)}$$
The second derivative [src]
  /  /                             2\                  \
  |  |                /       2   \ |     /       2   \|
  |  |         2      \1 + cot (x)/ |   2*\1 + cot (x)/|
2*|x*|2 + 2*cot (x) - --------------| - ---------------|
  |  |                      2       |        cot(x)    |
  \  \                   cot (x)    /                  /
$$2 \left(x \left(- \frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + 2 \cot^{2}{\left(x \right)} + 2\right) - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}}\right)$$
The third derivative [src]
  /                               2                     /                        2                  \\
  |                  /       2   \                      |           /       2   \      /       2   \||
  |         2      3*\1 + cot (x)/        /       2   \ |           \1 + cot (x)/    2*\1 + cot (x)/||
2*|6 + 6*cot (x) - ---------------- - 2*x*\1 + cot (x)/*|2*cot(x) + -------------- - ---------------||
  |                       2                             |                 3               cot(x)    ||
  \                    cot (x)                          \              cot (x)                      //
$$2 \left(- 2 x \left(\cot^{2}{\left(x \right)} + 1\right) \left(\frac{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{3}{\left(x \right)}} - \frac{2 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot{\left(x \right)}} + 2 \cot{\left(x \right)}\right) - \frac{3 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{2}{\left(x \right)}} + 6 \cot^{2}{\left(x \right)} + 6\right)$$