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Identical expressions
sqrt(twenty-seven +6x-(x^ two))
square root of (27 plus 6x minus (x squared ))
square root of (twenty minus seven plus 6x minus (x to the power of two))
√(27+6x-(x^2))
sqrt(27+6x-(x2))
sqrt27+6x-x2
sqrt(27+6x-(x²))
sqrt(27+6x-(x to the power of 2))
sqrt27+6x-x^2
Similar expressions
sqrt(27+6x+(x^2))
sqrt(27-6x-(x^2))

The derivative of the function/ sqrt(27+6x-(x^2))

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

The solution

You have entered [src]

   _______________
  /             2 
\/  27 + 6*x - x

$$\sqrt{- x^{2} + 6 x + 27}$$

  /   _______________\
d |  /             2 |
--\\/  27 + 6*x - x  /
dx

$$\frac{d}{d x} \sqrt{- x^{2} + 6 x + 27}$$

Transform

Detail solution

Let $u = - x^{2} + 6 x + 27$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x^{2} + 6 x + 27\right)$ :
1. Differentiate $- x^{2} + 6 x + 27$ term by term:
  1. The derivative of the constant $27$ 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: $x$ goes to $1$
    So, the result is: $6$
  3. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $- 2 x$
  The result is: $6 - 2 x$
The result of the chain rule is:

$\frac{6 - 2 x}{2 \sqrt{- x^{2} + 6 x + 27}}$
Now simplify:

$\frac{3 - x}{\sqrt{- x^{2} + 6 x + 27}}$

The answer is:

$\frac{3 - x}{\sqrt{- x^{2} + 6 x + 27}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      3 - x       
------------------
   _______________
  /             2 
\/  27 + 6*x - x

$$\frac{- x + 3}{\sqrt{- x^{2} + 6 x + 27}}$$

Simplify

The second derivative [src]

 /              2  \ 
 |      (-3 + x)   | 
-|1 + -------------| 
 |          2      | 
 \    27 - x  + 6*x/ 
---------------------
     _______________ 
    /       2        
  \/  27 - x  + 6*x

$$- \frac{\frac{\left(x - 3\right)^{2}}{- x^{2} + 6 x + 27} + 1}{\sqrt{- x^{2} + 6 x + 27}}$$

Simplify

The third derivative [src]

   /              2  \         
   |      (-3 + x)   |         
-3*|1 + -------------|*(-3 + x)
   |          2      |         
   \    27 - x  + 6*x/         
-------------------------------
                      3/2      
       /      2      \         
       \27 - x  + 6*x/

$$- \frac{3 \left(x - 3\right) \left(\frac{\left(x - 3\right)^{2}}{- x^{2} + 6 x + 27} + 1\right)}{\left(- x^{2} + 6 x + 27\right)^{\frac{3}{2}}}$$

Simplify

The graph

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

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The solution