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Derivative of (x^2-x)*(x^3+x)
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Identical expressions
sqrt(x^ two -6x- eleven)
square root of (x squared minus 6x minus 11)
square root of (x to the power of two minus 6x minus eleven)
√(x^2-6x-11)
sqrt(x2-6x-11)
sqrtx2-6x-11
sqrt(x²-6x-11)
sqrt(x to the power of 2-6x-11)
sqrtx^2-6x-11
Similar expressions
sqrt(x^2-6x+11)
sqrt(x^2+6x-11)

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

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

The solution

You have entered [src]

   _______________
  /  2            
\/  x  - 6*x - 11

$$\sqrt{x^{2} - 6 x - 11}$$

  /   _______________\
d |  /  2            |
--\\/  x  - 6*x - 11 /
dx

$$\frac{d}{d x} \sqrt{x^{2} - 6 x - 11}$$

Transform

Detail solution

Let $u = x^{2} - 6 x - 11$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - 6 x - 11\right)$ :
1. Differentiate $x^{2} - 6 x - 11$ 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: $6$
    So, the result is: $-6$
  3. The derivative of the constant $\left(-1\right) 11$ is zero.
  The result is: $2 x - 6$
The result of the chain rule is:

$\frac{2 x - 6}{2 \sqrt{x^{2} - 6 x - 11}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      -3 + x      
------------------
   _______________
  /  2            
\/  x  - 6*x - 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

$$\frac{- \frac{\left(x - 3\right)^{2}}{x^{2} - 6 x - 11} + 1}{\sqrt{x^{2} - 6 x - 11}}$$

Simplify

The third derivative [src]

  /               2   \         
  |       (-3 + x)    |         
3*|-1 + --------------|*(-3 + x)
  |            2      |         
  \     -11 + x  - 6*x/         
--------------------------------
                      3/2       
      /       2      \          
      \-11 + x  - 6*x/

$$\frac{3 \left(x - 3\right) \left(\frac{\left(x - 3\right)^{2}}{x^{2} - 6 x - 11} - 1\right)}{\left(x^{2} - 6 x - 11\right)^{\frac{3}{2}}}$$

Simplify

The graph

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

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The solution