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sqrt(x^2-12x+56)
The derivative of the function/ sqrt(x^2-12x+56)

Derivative of sqrt(x^2-12x+56)

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

You have entered [src] 
   ________________
  /  2             
\/  x  - 12*x + 56
x2−12x+56\sqrt{x^{2} - 12 x + 56} 
  /   ________________\
d |  /  2             |
--\\/  x  - 12*x + 56 /
dx
ddxx2−12x+56\frac{d}{d x} \sqrt{x^{2} - 12 x + 56}
Transform
Detail solution

  1. Let u=x2−12x+56u = x^{2} - 12 x + 56.

  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(x2−12x+56)\frac{d}{d x} \left(x^{2} - 12 x + 56\right):

    1. Differentiate x2−12x+56x^{2} - 12 x + 56 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: 1212

        So, the result is: −12-12

      3. The derivative of the constant 5656 is zero.

      The result is: 2x−122 x - 12

    The result of the chain rule is:

    2x−122x2−12x+56\frac{2 x - 12}{2 \sqrt{x^{2} - 12 x + 56}}

  4. Now simplify:

    x−6x2−12x+56\frac{x - 6}{\sqrt{x^{2} - 12 x + 56}}

The answer is:

x−6x2−12x+56\frac{x - 6}{\sqrt{x^{2} - 12 x + 56}}

The graph
02468-8-6-4-2-1010-2020
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       -6 + x      
-------------------
   ________________
  /  2             
\/  x  - 12*x + 56
x−6x2−12x+56\frac{x - 6}{\sqrt{x^{2} - 12 x + 56}}
Simplify
The second derivative [src] 
               2   
       (-6 + x)    
 1 - --------------
           2       
     56 + x  - 12*x
-------------------
   ________________
  /       2        
\/  56 + x  - 12*x
−(x−6)2x2−12x+56+1x2−12x+56\frac{- \frac{\left(x - 6\right)^{2}}{x^{2} - 12 x + 56} + 1}{\sqrt{x^{2} - 12 x + 56}}
Simplify
The third derivative [src] 
  /               2   \         
  |       (-6 + x)    |         
3*|-1 + --------------|*(-6 + x)
  |           2       |         
  \     56 + x  - 12*x/         
--------------------------------
                      3/2       
      /      2       \          
      \56 + x  - 12*x/
3(x−6)((x−6)2x2−12x+56−1)(x2−12x+56)32\frac{3 \left(x - 6\right) \left(\frac{\left(x - 6\right)^{2}}{x^{2} - 12 x + 56} - 1\right)}{\left(x^{2} - 12 x + 56\right)^{\frac{3}{2}}}
Simplify
The graph
Derivative of sqrt(x^2-12x+56)
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