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y=1/ctg^2x+1/3(ctgx)

Derivative of y=1/ctg^2x+1/3(ctgx)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
     1      cot(x)
1*------- + ------
     2        3   
  cot (x)         
$$\frac{\cot{\left(x \right)}}{3} + 1 \cdot \frac{1}{\cot^{2}{\left(x \right)}}$$
d /     1      cot(x)\
--|1*------- + ------|
dx|     2        3   |
  \  cot (x)         /
$$\frac{d}{d x} \left(\frac{\cot{\left(x \right)}}{3} + 1 \cdot \frac{1}{\cot^{2}{\left(x \right)}}\right)$$
Detail solution
  1. Differentiate term by term:

    1. Apply the quotient rule, which is:

      and .

      To find :

      1. The derivative of the constant is zero.

      To find :

      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:

      Now plug in to the quotient rule:

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

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
         2                2   
  1   cot (x)   -2 - 2*cot (x)
- - - ------- - --------------
  3      3                2   
                cot(x)*cot (x)
$$- \frac{- 2 \cot^{2}{\left(x \right)} - 2}{\cot{\left(x \right)} \cot^{2}{\left(x \right)}} - \frac{\cot^{2}{\left(x \right)}}{3} - \frac{1}{3}$$
The second derivative [src]
  /       2   \ /              /       2   \         \
  |1   cot (x)| |     6      9*\1 + cot (x)/         |
2*|- + -------|*|- ------- + --------------- + cot(x)|
  \3      3   / |     2             4                |
                \  cot (x)       cot (x)             /
$$2 \left(\frac{\cot^{2}{\left(x \right)}}{3} + \frac{1}{3}\right) \left(\frac{9 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot^{4}{\left(x \right)}} + \cot{\left(x \right)} - \frac{6}{\cot^{2}{\left(x \right)}}\right)$$
The third derivative [src]
                /                                                             2\
  /       2   \ |                             /       2   \      /       2   \ |
  |1   cot (x)| |          2        12     48*\1 + cot (x)/   36*\1 + cot (x)/ |
2*|- + -------|*|-1 - 3*cot (x) + ------ - ---------------- + -----------------|
  \3      3   / |                 cot(x)          3                   5        |
                \                              cot (x)             cot (x)     /
$$2 \left(\frac{\cot^{2}{\left(x \right)}}{3} + \frac{1}{3}\right) \left(\frac{36 \left(\cot^{2}{\left(x \right)} + 1\right)^{2}}{\cot^{5}{\left(x \right)}} - \frac{48 \left(\cot^{2}{\left(x \right)} + 1\right)}{\cot^{3}{\left(x \right)}} - 3 \cot^{2}{\left(x \right)} - 1 + \frac{12}{\cot{\left(x \right)}}\right)$$
The graph
Derivative of y=1/ctg^2x+1/3(ctgx)