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Derivative of:
Derivative of 6^x
Derivative of (x^2-1)/(x^2+1)
Derivative of (x+4)^2
Derivative of (x+3)/(x-1)
Identical expressions
sqrt(x^ two -12x+ fifty-six)
square root of (x squared minus 12x plus 56)
square root of (x to the power of two minus 12x plus fifty minus six)
√(x^2-12x+56)
sqrt(x2-12x+56)
sqrtx2-12x+56
sqrt(x²-12x+56)
sqrt(x to the power of 2-12x+56)
sqrtx^2-12x+56
Similar expressions
sqrt(x^2+12x+56)
sqrt(x^2-12x-56)

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

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

The solution

You have entered [src]

   ________________
  /  2             
\/  x  - 12*x + 56

$$\sqrt{x^{2} - 12 x + 56}$$

  /   ________________\
d |  /  2             |
--\\/  x  - 12*x + 56 /
dx

$$\frac{d}{d x} \sqrt{x^{2} - 12 x + 56}$$

Transform

Detail solution

Let $u = x^{2} - 12 x + 56$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - 12 x + 56\right)$ :
1. Differentiate $x^{2} - 12 x + 56$ term by term:
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  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: $x$ goes to $1$
      So, the result is: $12$
    So, the result is: $-12$
  3. The derivative of the constant $56$ is zero.
  The result is: $2 x - 12$
The result of the chain rule is:

$\frac{2 x - 12}{2 \sqrt{x^{2} - 12 x + 56}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       -6 + x      
-------------------
   ________________
  /  2             
\/  x  - 12*x + 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

$$\frac{- \frac{\left(x - 6\right)^{2}}{x^{2} - 12 x + 56} + 1}{\sqrt{x^{2} - 12 x + 56}}$$

Simplify

The third derivative [src]

  /               2   \         
  |       (-6 + x)    |         
3*|-1 + --------------|*(-6 + x)
  |           2       |         
  \     56 + x  - 12*x/         
--------------------------------
                      3/2       
      /      2       \          
      \56 + x  - 12*x/

$$\frac{3 \left(x - 6\right) \left(\frac{\left(x - 6\right)^{2}}{x^{2} - 12 x + 56} - 1\right)}{\left(x^{2} - 12 x + 56\right)^{\frac{3}{2}}}$$

Simplify

The graph

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

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The solution