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sqrt(x^2-6x-11)
The derivative of the function/ sqrt(x^2-6x-11)

Derivative of sqrt(x^2-6x-11)

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

You have entered [src] 
   _______________
  /  2            
\/  x  - 6*x - 11
x2−6x−11\sqrt{x^{2} - 6 x - 11} 
  /   _______________\
d |  /  2            |
--\\/  x  - 6*x - 11 /
dx
ddxx2−6x−11\frac{d}{d x} \sqrt{x^{2} - 6 x - 11}
Transform
Detail solution

  1. Let u=x2−6x−11u = x^{2} - 6 x - 11.

  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−11)\frac{d}{d x} \left(x^{2} - 6 x - 11\right):

    1. Differentiate x2−6x−11x^{2} - 6 x - 11 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. 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: 66

        So, the result is: −6-6

      3. The derivative of the constant (−1)11\left(-1\right) 11 is zero.

      The result is: 2x−62 x - 6

    The result of the chain rule is:

    2x−62x2−6x−11\frac{2 x - 6}{2 \sqrt{x^{2} - 6 x - 11}}

  4. Now simplify:

    x−3x2−6x−11\frac{x - 3}{\sqrt{x^{2} - 6 x - 11}}

The answer is:

x−3x2−6x−11\frac{x - 3}{\sqrt{x^{2} - 6 x - 11}}

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 - 11
x−3x2−6x−11\frac{x - 3}{\sqrt{x^{2} - 6 x - 11}}
Simplify
The second derivative [src] 
               2   
       (-3 + x)    
 1 - --------------
            2      
     -11 + x  - 6*x
-------------------
   ________________
  /        2       
\/  -11 + x  - 6*x
−(x−3)2x2−6x−11+1x2−6x−11\frac{- \frac{\left(x - 3\right)^{2}}{x^{2} - 6 x - 11} + 1}{\sqrt{x^{2} - 6 x - 11}}
Simplify
The third derivative [src] 
  /               2   \         
  |       (-3 + x)    |         
3*|-1 + --------------|*(-3 + x)
  |            2      |         
  \     -11 + x  - 6*x/         
--------------------------------
                      3/2       
      /       2      \          
      \-11 + x  - 6*x/
3(x−3)((x−3)2x2−6x−11−1)(x2−6x−11)32\frac{3 \left(x - 3\right) \left(\frac{\left(x - 3\right)^{2}}{x^{2} - 6 x - 11} - 1\right)}{\left(x^{2} - 6 x - 11\right)^{\frac{3}{2}}}
Simplify
The graph
Derivative of sqrt(x^2-6x-11)
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