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sqrt(x^2-6*x+13)
The derivative of the function/ sqrt(x^2-6*x+13)

Derivative of sqrt(x^2-6*x+13)

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

You have entered [src] 
   _______________
  /  2            
\/  x  - 6*x + 13
(x2−6x)+13\sqrt{\left(x^{2} - 6 x\right) + 13} 
sqrt(x^2 - 6*x + 13)
Detail solution

  1. Let u=(x2−6x)+13u = \left(x^{2} - 6 x\right) + 13.

  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((x2−6x)+13)\frac{d}{d x} \left(\left(x^{2} - 6 x\right) + 13\right):

    1. Differentiate (x2−6x)+13\left(x^{2} - 6 x\right) + 13 term by term:

      1. Differentiate x2−6xx^{2} - 6 x term by term:

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

        2. 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: −6-6

        The result is: 2x−62 x - 6

      2. The derivative of the constant 1313 is zero.

      The result is: 2x−62 x - 6

    The result of the chain rule is:

    2x−62(x2−6x)+13\frac{2 x - 6}{2 \sqrt{\left(x^{2} - 6 x\right) + 13}}

  4. Now simplify:

    x−3x2−6x+13\frac{x - 3}{\sqrt{x^{2} - 6 x + 13}}

The answer is:

x−3x2−6x+13\frac{x - 3}{\sqrt{x^{2} - 6 x + 13}}

The graph
02468-8-6-4-2-1010-2020
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
      -3 + x      
------------------
   _______________
  /  2            
\/  x  - 6*x + 13
x−3(x2−6x)+13\frac{x - 3}{\sqrt{\left(x^{2} - 6 x\right) + 13}}
Simplify
The second derivative [src] 
              2   
      (-3 + x)    
1 - ------------- 
          2       
    13 + x  - 6*x 
------------------
   _______________
  /       2       
\/  13 + x  - 6*x
−(x−3)2x2−6x+13+1x2−6x+13\frac{- \frac{\left(x - 3\right)^{2}}{x^{2} - 6 x + 13} + 1}{\sqrt{x^{2} - 6 x + 13}}
Simplify
The third derivative [src] 
  /               2  \         
  |       (-3 + x)   |         
3*|-1 + -------------|*(-3 + x)
  |           2      |         
  \     13 + x  - 6*x/         
-------------------------------
                      3/2      
       /      2      \         
       \13 + x  - 6*x/
3(x−3)((x−3)2x2−6x+13−1)(x2−6x+13)32\frac{3 \left(x - 3\right) \left(\frac{\left(x - 3\right)^{2}}{x^{2} - 6 x + 13} - 1\right)}{\left(x^{2} - 6 x + 13\right)^{\frac{3}{2}}}
Simplify
The graph
Derivative of sqrt(x^2-6*x+13)
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