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ctg(arcsin(sqrt(-2*x+2)))

Derivative of ctg(arcsin(sqrt(-2*x+2)))

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

from to

Piecewise:

The solution

You have entered [src]
   /    /  __________\\
cot\asin\\/ -2*x + 2 //
$$\cot{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)}$$
d /   /    /  __________\\\
--\cot\asin\\/ -2*x + 2 ///
dx                         
$$\frac{d}{d x} \cot{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)}$$
Detail solution
  1. There are multiple ways to do this derivative.

    Method #1

    1. Rewrite the function to be differentiated:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. Let .

      2. Apply the power rule: goes to

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

        1. Differentiate term by term:

          1. The derivative of the constant is zero.

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

            1. Apply the power rule: goes to

            So, the result is:

          The result is:

        The result of the chain rule is:

      To find :

      1. Let .

      2. Apply the power rule: goes to

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

        1. Differentiate term by term:

          1. The derivative of the constant is zero.

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

            1. Apply the power rule: goes to

            So, the result is:

          The result is:

        The result of the chain rule is:

      Now plug in to the quotient rule:

    Method #2

    1. Rewrite the function to be differentiated:

    2. Apply the quotient rule, which is:

      and .

      To find :

      1. Let .

      2. Apply the power rule: goes to

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

        1. Differentiate term by term:

          1. The derivative of the constant is zero.

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

            1. Apply the power rule: goes to

            So, the result is:

          The result is:

        The result of the chain rule is:

      To find :

      1. Let .

      2. Apply the power rule: goes to

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

        1. Differentiate term by term:

          1. The derivative of the constant is zero.

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

            1. Apply the power rule: goes to

            So, the result is:

          The result is:

        The result of the chain rule is:

      Now plug in to the quotient rule:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
 /        2/    /  __________\\\ 
-\-1 - cot \asin\\/ -2*x + 2 /// 
---------------------------------
      __________   __________    
    \/ -1 + 2*x *\/ -2*x + 2     
$$- \frac{- \cot^{2}{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)} - 1}{\sqrt{2 - 2 x} \sqrt{2 x - 1}}$$
The second derivative [src]
  ___ /    -1 + 2*x\ /    1         1         1    \
\/ 2 *|2 - --------|*|--------- - ------ - --------|
      \     -1 + x / \2*(1 - x)   -1 + x   -1 + 2*x/
----------------------------------------------------
                  _______   __________              
              4*\/ 1 - x *\/ -1 + 2*x               
$$\frac{\sqrt{2} \cdot \left(2 - \frac{2 x - 1}{x - 1}\right) \left(- \frac{1}{2 x - 1} - \frac{1}{x - 1} + \frac{1}{2 \cdot \left(1 - x\right)}\right)}{4 \sqrt{1 - x} \sqrt{2 x - 1}}$$
The third derivative [src]
                     /                                                                                         -1 + 2*x   \
                     |                                                                                     2 - --------   |
  ___ /    -1 + 2*x\ |    3            6            7                2                     6                    -1 + x    |
\/ 2 *|2 - --------|*|--------- + ----------- + ---------- - ------------------ + ------------------- + ------------------|
      \     -1 + x / |        2             2            2   (1 - x)*(-1 + 2*x)   (-1 + x)*(-1 + 2*x)   (1 - x)*(-1 + 2*x)|
                     \(-1 + x)    (-1 + 2*x)    2*(1 - x)                                                                 /
---------------------------------------------------------------------------------------------------------------------------
                                                      _______   __________                                                 
                                                  8*\/ 1 - x *\/ -1 + 2*x                                                  
$$\frac{\sqrt{2} \cdot \left(2 - \frac{2 x - 1}{x - 1}\right) \left(\frac{6}{\left(2 x - 1\right)^{2}} + \frac{6}{\left(x - 1\right) \left(2 x - 1\right)} + \frac{3}{\left(x - 1\right)^{2}} + \frac{2 - \frac{2 x - 1}{x - 1}}{\left(1 - x\right) \left(2 x - 1\right)} - \frac{2}{\left(1 - x\right) \left(2 x - 1\right)} + \frac{7}{2 \left(1 - x\right)^{2}}\right)}{8 \sqrt{1 - x} \sqrt{2 x - 1}}$$
The graph
Derivative of ctg(arcsin(sqrt(-2*x+2)))