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x-sqrt(x^2-x)

Derivative of x-sqrt(x^2-x)

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

The graph:

from to

Piecewise:

The solution

You have entered [src]
       ________
      /  2     
x - \/  x  - x 
$$x - \sqrt{x^{2} - x}$$
x - sqrt(x^2 - x)
Detail solution
  1. Differentiate term by term:

    1. Apply the power rule: goes to

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

      1. Let .

      2. Apply the power rule: goes to

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

        1. Differentiate term by term:

          1. Apply the power rule: goes to

          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:

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
      -1/2 + x 
1 - -----------
       ________
      /  2     
    \/  x  - x 
$$- \frac{x - \frac{1}{2}}{\sqrt{x^{2} - x}} + 1$$
The second derivative [src]
               2 
     (-1 + 2*x)  
-1 + ------------
     4*x*(-1 + x)
-----------------
    ____________ 
  \/ x*(-1 + x)  
$$\frac{-1 + \frac{\left(2 x - 1\right)^{2}}{4 x \left(x - 1\right)}}{\sqrt{x \left(x - 1\right)}}$$
The third derivative [src]
             /              2\
             |    (-1 + 2*x) |
3*(-1 + 2*x)*|4 - -----------|
             \     x*(-1 + x)/
------------------------------
                    3/2       
      8*(x*(-1 + x))          
$$\frac{3 \left(4 - \frac{\left(2 x - 1\right)^{2}}{x \left(x - 1\right)}\right) \left(2 x - 1\right)}{8 \left(x \left(x - 1\right)\right)^{\frac{3}{2}}}$$
The graph
Derivative of x-sqrt(x^2-x)