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sqrt(27+6x-(x^2))

Derivative of sqrt(27+6x-(x^2))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   _______________
  /             2 
\/  27 + 6*x - x  
$$\sqrt{- x^{2} + 6 x + 27}$$
  /   _______________\
d |  /             2 |
--\\/  27 + 6*x - x  /
dx                    
$$\frac{d}{d x} \sqrt{- x^{2} + 6 x + 27}$$
Detail solution
  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. The derivative of the constant is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      3. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: goes to

        So, the result is:

      The result is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
      3 - x       
------------------
   _______________
  /             2 
\/  27 + 6*x - x  
$$\frac{- x + 3}{\sqrt{- x^{2} + 6 x + 27}}$$
The second derivative [src]
 /              2  \ 
 |      (-3 + x)   | 
-|1 + -------------| 
 |          2      | 
 \    27 - x  + 6*x/ 
---------------------
     _______________ 
    /       2        
  \/  27 - x  + 6*x  
$$- \frac{\frac{\left(x - 3\right)^{2}}{- x^{2} + 6 x + 27} + 1}{\sqrt{- x^{2} + 6 x + 27}}$$
The third derivative [src]
   /              2  \         
   |      (-3 + x)   |         
-3*|1 + -------------|*(-3 + x)
   |          2      |         
   \    27 - x  + 6*x/         
-------------------------------
                      3/2      
       /      2      \         
       \27 - x  + 6*x/         
$$- \frac{3 \left(x - 3\right) \left(\frac{\left(x - 3\right)^{2}}{- x^{2} + 6 x + 27} + 1\right)}{\left(- x^{2} + 6 x + 27\right)^{\frac{3}{2}}}$$
The graph
Derivative of sqrt(27+6x-(x^2))