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

Derivative of sqrt(2x-x^2)

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

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The solution

You have entered [src] 
   __________
  /        2 
\/  2*x - x
−x2+2x\sqrt{- x^{2} + 2 x} 
  /   __________\
d |  /        2 |
--\\/  2*x - x  /
dx
ddx−x2+2x\frac{d}{d x} \sqrt{- x^{2} + 2 x}
Transform
Detail solution

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

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

      1. 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

      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: 2−2x2 - 2 x

    The result of the chain rule is:

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

  4. Now simplify:

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

The answer is:

1−xx(2−x)\frac{1 - x}{\sqrt{x \left(2 - 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 - x    
-------------
   __________
  /        2 
\/  2*x - x
1−x−x2+2x\frac{1 - x}{\sqrt{- x^{2} + 2 x}}
Simplify
The second derivative [src] 
 /            2\ 
 |    (-1 + x) | 
-|1 + ---------| 
 \    x*(2 - x)/ 
-----------------
    ___________  
  \/ x*(2 - x)
−1+(x−1)2x(2−x)x(2−x)- \frac{1 + \frac{\left(x - 1\right)^{2}}{x \left(2 - x\right)}}{\sqrt{x \left(2 - x\right)}}
Simplify
The third derivative [src] 
   /            2\         
   |    (-1 + x) |         
-3*|1 + ---------|*(-1 + x)
   \    x*(2 - x)/         
---------------------------
                  3/2      
       (x*(2 - x))
−3⋅(1+(x−1)2x(2−x))(x−1)(x(2−x))32- \frac{3 \cdot \left(1 + \frac{\left(x - 1\right)^{2}}{x \left(2 - x\right)}\right) \left(x - 1\right)}{\left(x \left(2 - x\right)\right)^{\frac{3}{2}}}
Simplify
The graph
Derivative of sqrt(2x-x^2)
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