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

Derivative of sqrt(x-x^2)

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

You have entered [src] 
   ________
  /      2 
\/  x - x
−x2+x\sqrt{- x^{2} + x} 
sqrt(x - x^2)
Detail solution

  1. Let u=−x2+xu = - x^{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(−x2+x)\frac{d}{d x} \left(- x^{2} + x\right):

    1. Differentiate −x2+x- x^{2} + x term by term:

      1. Apply the power rule: xx goes to 11

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

        1. Apply the power rule: x2x^{2} goes to 2x2 x

        So, the result is: −2x- 2 x

      The result is: 1−2x1 - 2 x

    The result of the chain rule is:

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

  4. Now simplify:

    12−xx(1−x)\frac{\frac{1}{2} - x}{\sqrt{x \left(1 - x\right)}}

The answer is:

12−xx(1−x)\frac{\frac{1}{2} - x}{\sqrt{x \left(1 - x\right)}}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  1/2 - x  
-----------
   ________
  /      2 
\/  x - x
12−x−x2+x\frac{\frac{1}{2} - x}{\sqrt{- x^{2} + x}}
Simplify
The second derivative [src] 
 /              2\ 
 |    (-1 + 2*x) | 
-|1 + -----------| 
 \    4*x*(1 - x)/ 
-------------------
     ___________   
   \/ x*(1 - x)
−1+(2x−1)24x(1−x)x(1−x)- \frac{1 + \frac{\left(2 x - 1\right)^{2}}{4 x \left(1 - x\right)}}{\sqrt{x \left(1 - x\right)}}
Simplify
3-я производная [src] 
              /              2\
              |    (-1 + 2*x) |
-3*(-1 + 2*x)*|4 + -----------|
              \     x*(1 - x) /
-------------------------------
                     3/2       
        8*(x*(1 - x))
−3(4+(2x−1)2x(1−x))(2x−1)8(x(1−x))32- \frac{3 \left(4 + \frac{\left(2 x - 1\right)^{2}}{x \left(1 - x\right)}\right) \left(2 x - 1\right)}{8 \left(x \left(1 - x\right)\right)^{\frac{3}{2}}}
Simplify
The third derivative [src] 
              /              2\
              |    (-1 + 2*x) |
-3*(-1 + 2*x)*|4 + -----------|
              \     x*(1 - x) /
-------------------------------
                     3/2       
        8*(x*(1 - x))
−3(4+(2x−1)2x(1−x))(2x−1)8(x(1−x))32- \frac{3 \left(4 + \frac{\left(2 x - 1\right)^{2}}{x \left(1 - x\right)}\right) \left(2 x - 1\right)}{8 \left(x \left(1 - x\right)\right)^{\frac{3}{2}}}
Simplify
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