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Derivative of:
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Graphing y =:
x-sqrt(x^2-x)
Limit of the function:
x-sqrt(x^2-x)
Identical expressions
x-sqrt(x^ two -x)
x minus square root of (x squared minus x)
x minus square root of (x to the power of two minus x)
x-√(x^2-x)
x-sqrt(x2-x)
x-sqrtx2-x
x-sqrt(x²-x)
x-sqrt(x to the power of 2-x)
x-sqrtx^2-x
Similar expressions
x+sqrt(x^2-x)
x-sqrt(x^2+x)

The derivative of the function/ x-sqrt(x^2-x)

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

The solution

You have entered [src]

       ________
      /  2     
x - \/  x  - x

$$x - \sqrt{x^{2} - x}$$

x - sqrt(x^2 - x)

Detail solution

Differentiate $x - \sqrt{x^{2} - x}$ term by term:
1. Apply the power rule: $x$ goes to $1$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = x^{2} - x$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - x\right)$ :
    1. Differentiate $x^{2} - x$ term by term:
      1. Apply the power rule: $x^{2}$ goes to $2 x$
      2. 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: $-1$
      The result is: $2 x - 1$
    The result of the chain rule is:
    
    $\frac{2 x - 1}{2 \sqrt{x^{2} - x}}$
  So, the result is: $- \frac{2 x - 1}{2 \sqrt{x^{2} - x}}$
The result is: $- \frac{2 x - 1}{2 \sqrt{x^{2} - x}} + 1$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      -1/2 + x 
1 - -----------
       ________
      /  2     
    \/  x  - x

$$- \frac{x - \frac{1}{2}}{\sqrt{x^{2} - x}} + 1$$

Simplify

The second derivative [src]

               2 
     (-1 + 2*x)  
-1 + ------------
     4*x*(-1 + x)
-----------------
    ____________ 
  \/ x*(-1 + x)

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

Simplify

The third derivative [src]

             /              2\
             |    (-1 + 2*x) |
3*(-1 + 2*x)*|4 - -----------|
             \     x*(-1 + x)/
------------------------------
                    3/2       
      8*(x*(-1 + x))

$$\frac{3 \left(4 - \frac{\left(2 x - 1\right)^{2}}{x \left(x - 1\right)}\right) \left(2 x - 1\right)}{8 \left(x \left(x - 1\right)\right)^{\frac{3}{2}}}$$

Simplify

The graph

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

The graph:

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The solution