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Derivative of:
Derivative of 4/x^3
Derivative of (x+3)^2*(x+5)-1
Derivative of x^3+2
Derivative of x^(-2)
Identical expressions
ctg(arcsin(sqrt(- two *x+ two)))
ctg(arc sinus of ( square root of ( minus 2 multiply by x plus 2)))
ctg(arc sinus of ( square root of ( minus two multiply by x plus two)))
ctg(arcsin(√(-2*x+2)))
ctg(arcsin(sqrt(-2x+2)))
ctgarcsinsqrt-2x+2
Similar expressions
ctg(arcsin(sqrt(2*x+2)))
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)))

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

$$\frac{d}{d x} \cot{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)}$$

Transform

Detail solution

There are multiple ways to do this derivative.
Method #1
1. Rewrite the function to be differentiated:
  
  $\cot{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)} = \frac{\sqrt{2 x - 1}}{\sqrt{2 - 2 x}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sqrt{2 x - 1}$ and $g{\left(x \right)} = \sqrt{2 - 2 x}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 2 x - 1$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 1\right)$ :
    1. Differentiate $2 x - 1$ term by term:
      
      The derivative of the constant $-1$ is zero.
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result is: $2$
    The result of the chain rule is:
    
    $\frac{1}{\sqrt{2 x - 1}}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2 - 2 x$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 - 2 x\right)$ :
    1. Differentiate $2 - 2 x$ term by term:
      
      The derivative of the constant $2$ is zero.
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $-2$
      
      The result is: $-2$
    The result of the chain rule is:
    
    $- \frac{1}{\sqrt{2 - 2 x}}$
  Now plug in to the quotient rule:
  
  $\frac{\frac{\sqrt{2 - 2 x}}{\sqrt{2 x - 1}} + \frac{\sqrt{2 x - 1}}{\sqrt{2 - 2 x}}}{2 - 2 x}$
Method #2
1. Rewrite the function to be differentiated:
  
  $\cot{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)} = \frac{\sqrt{2 x - 1}}{\sqrt{2 - 2 x}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sqrt{2 x - 1}$ and $g{\left(x \right)} = \sqrt{2 - 2 x}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 2 x - 1$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 1\right)$ :
    1. Differentiate $2 x - 1$ term by term:
      
      The derivative of the constant $-1$ is zero.
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result is: $2$
    The result of the chain rule is:
    
    $\frac{1}{\sqrt{2 x - 1}}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2 - 2 x$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 - 2 x\right)$ :
    1. Differentiate $2 - 2 x$ term by term:
      
      The derivative of the constant $2$ is zero.
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $-2$
      
      The result is: $-2$
    The result of the chain rule is:
    
    $- \frac{1}{\sqrt{2 - 2 x}}$
  Now plug in to the quotient rule:
  
  $\frac{\frac{\sqrt{2 - 2 x}}{\sqrt{2 x - 1}} + \frac{\sqrt{2 x - 1}}{\sqrt{2 - 2 x}}}{2 - 2 x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 /        2/    /  __________\\\ 
-\-1 - cot \asin\\/ -2*x + 2 /// 
---------------------------------
      __________   __________    
    \/ -1 + 2*x *\/ -2*x + 2

$$- \frac{- \cot^{2}{\left(\operatorname{asin}{\left(\sqrt{2 - 2 x} \right)} \right)} - 1}{\sqrt{2 - 2 x} \sqrt{2 x - 1}}$$

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

$$\frac{\sqrt{2} \cdot \left(2 - \frac{2 x - 1}{x - 1}\right) \left(- \frac{1}{2 x - 1} - \frac{1}{x - 1} + \frac{1}{2 \cdot \left(1 - x\right)}\right)}{4 \sqrt{1 - x} \sqrt{2 x - 1}}$$

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

$$\frac{\sqrt{2} \cdot \left(2 - \frac{2 x - 1}{x - 1}\right) \left(\frac{6}{\left(2 x - 1\right)^{2}} + \frac{6}{\left(x - 1\right) \left(2 x - 1\right)} + \frac{3}{\left(x - 1\right)^{2}} + \frac{2 - \frac{2 x - 1}{x - 1}}{\left(1 - x\right) \left(2 x - 1\right)} - \frac{2}{\left(1 - x\right) \left(2 x - 1\right)} + \frac{7}{2 \left(1 - x\right)^{2}}\right)}{8 \sqrt{1 - x} \sqrt{2 x - 1}}$$

Simplify

The graph

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

The graph:

Piecewise:

The solution

Method #1

Method #2