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ctg(arcsin(sqrt(-2*x+2)))
The derivative of the function/ ctg(arcsin(sqrt(-2*x+2)))

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

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You have entered [src] 
   /    /  __________\\
cot\asin\\/ -2*x + 2 //
cot⁡(asin⁡(2−2x))\cot{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)} 
d /   /    /  __________\\\
--\cot\asin\\/ -2*x + 2 ///
dx
ddxcot⁡(asin⁡(2−2x))\frac{d}{d x} \cot{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)}
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Detail solution

  1. There are multiple ways to do this derivative.

    Method #1

    1. Rewrite the function to be differentiated:

      cot⁡(asin⁡(2−2x))=2x−12−2x\cot{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)} = \frac{\sqrt{2 x - 1}}{\sqrt{2 - 2 x}}

    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)=2x−1f{\left(x \right)} = \sqrt{2 x - 1} and g(x)=2−2xg{\left(x \right)} = \sqrt{2 - 2 x}.

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

      1. Let u=2x−1u = 2 x - 1.

      2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

      3. Then, apply the chain rule. Multiply by ddx(2x−1)\frac{d}{d x} \left(2 x - 1\right):

        1. Differentiate 2x−12 x - 1 term by term:

          1. The derivative of the constant −1-1 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: xx goes to 11

            So, the result is: 22

          The result is: 22

        The result of the chain rule is:

        12x−1\frac{1}{\sqrt{2 x - 1}}

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

      1. Let u=2−2xu = 2 - 2 x.

      2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

      3. Then, apply the chain rule. Multiply by ddx(2−2x)\frac{d}{d x} \left(2 - 2 x\right):

        1. Differentiate 2−2x2 - 2 x term by term:

          1. The derivative of the constant 22 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: xx goes to 11

            So, the result is: −2-2

          The result is: −2-2

        The result of the chain rule is:

        −12−2x- \frac{1}{\sqrt{2 - 2 x}}

      Now plug in to the quotient rule:

      2−2x2x−1+2x−12−2x2−2x\frac{\frac{\sqrt{2 - 2 x}}{\sqrt{2 x - 1}} + \frac{\sqrt{2 x - 1}}{\sqrt{2 - 2 x}}}{2 - 2 x}

    Method #2

    1. Rewrite the function to be differentiated:

      cot⁡(asin⁡(2−2x))=2x−12−2x\cot{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)} = \frac{\sqrt{2 x - 1}}{\sqrt{2 - 2 x}}

    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)=2x−1f{\left(x \right)} = \sqrt{2 x - 1} and g(x)=2−2xg{\left(x \right)} = \sqrt{2 - 2 x}.

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

      1. Let u=2x−1u = 2 x - 1.

      2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

      3. Then, apply the chain rule. Multiply by ddx(2x−1)\frac{d}{d x} \left(2 x - 1\right):

        1. Differentiate 2x−12 x - 1 term by term:

          1. The derivative of the constant −1-1 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: xx goes to 11

            So, the result is: 22

          The result is: 22

        The result of the chain rule is:

        12x−1\frac{1}{\sqrt{2 x - 1}}

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

      1. Let u=2−2xu = 2 - 2 x.

      2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

      3. Then, apply the chain rule. Multiply by ddx(2−2x)\frac{d}{d x} \left(2 - 2 x\right):

        1. Differentiate 2−2x2 - 2 x term by term:

          1. The derivative of the constant 22 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: xx goes to 11

            So, the result is: −2-2

          The result is: −2-2

        The result of the chain rule is:

        −12−2x- \frac{1}{\sqrt{2 - 2 x}}

      Now plug in to the quotient rule:

      2−2x2x−1+2x−12−2x2−2x\frac{\frac{\sqrt{2 - 2 x}}{\sqrt{2 x - 1}} + \frac{\sqrt{2 x - 1}}{\sqrt{2 - 2 x}}}{2 - 2 x}

  2. Now simplify:

    24(1−x)322x−1\frac{\sqrt{2}}{4 \left(1 - x\right)^{\frac{3}{2}} \sqrt{2 x - 1}}

The answer is:

24(1−x)322x−1\frac{\sqrt{2}}{4 \left(1 - x\right)^{\frac{3}{2}} \sqrt{2 x - 1}}

The graph
02468-8-6-4-2-1010020
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The first derivative [src] 
 /        2/    /  __________\\\ 
-\-1 - cot \asin\\/ -2*x + 2 /// 
---------------------------------
      __________   __________    
    \/ -1 + 2*x *\/ -2*x + 2
−−cot⁡2(asin⁡(2−2x))−12−2x2x−1- \frac{- \cot^{2}{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)} - 1}{\sqrt{2 - 2 x} \sqrt{2 x - 1}}
Simplify
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
2⋅(2−2x−1x−1)(−12x−1−1x−1+12⋅(1−x))41−x2x−1\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}}
Simplify
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
2⋅(2−2x−1x−1)(6(2x−1)2+6(x−1)(2x−1)+3(x−1)2+2−2x−1x−1(1−x)(2x−1)−2(1−x)(2x−1)+72(1−x)2)81−x2x−1\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}}
Simplify
The graph
Derivative of ctg(arcsin(sqrt(-2*x+2)))
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