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Derivative of 8*x
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Graphing y =:
sqrt(9+8x-x^2)
Identical expressions
sqrt(nine +8x-x^ two)
square root of (9 plus 8x minus x squared )
square root of (nine plus 8x minus x to the power of two)
√(9+8x-x^2)
sqrt(9+8x-x2)
sqrt9+8x-x2
sqrt(9+8x-x²)
sqrt(9+8x-x to the power of 2)
sqrt9+8x-x^2
Similar expressions
sqrt(9-8x-x^2)
sqrt(9+8x+x^2)

The derivative of the function/ sqrt(9+8x-x^2)

Derivative of sqrt(9+8x-x^2)

The solution

You have entered [src]

   ______________
  /            2 
\/  9 + 8*x - x

$$\sqrt{- x^{2} + 8 x + 9}$$

  /   ______________\
d |  /            2 |
--\\/  9 + 8*x - x  /
dx

$$\frac{d}{d x} \sqrt{- x^{2} + 8 x + 9}$$

Transform

Detail solution

Let $u = - x^{2} + 8 x + 9$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x^{2} + 8 x + 9\right)$ :
1. Differentiate $- x^{2} + 8 x + 9$ term by term:
  1. The derivative of the constant $9$ 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: $8$
  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: $8 - 2 x$
The result of the chain rule is:

$\frac{8 - 2 x}{2 \sqrt{- x^{2} + 8 x + 9}}$
Now simplify:

$\frac{4 - x}{\sqrt{- x^{2} + 8 x + 9}}$

The answer is:

$\frac{4 - x}{\sqrt{- x^{2} + 8 x + 9}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      4 - x      
-----------------
   ______________
  /            2 
\/  9 + 8*x - x

$$\frac{- x + 4}{\sqrt{- x^{2} + 8 x + 9}}$$

Simplify

The second derivative [src]

 /             2  \ 
 |     (-4 + x)   | 
-|1 + ------------| 
 |         2      | 
 \    9 - x  + 8*x/ 
--------------------
    ______________  
   /      2         
 \/  9 - x  + 8*x

$$- \frac{\frac{\left(x - 4\right)^{2}}{- x^{2} + 8 x + 9} + 1}{\sqrt{- x^{2} + 8 x + 9}}$$

Simplify

The third derivative [src]

   /             2  \         
   |     (-4 + x)   |         
-3*|1 + ------------|*(-4 + x)
   |         2      |         
   \    9 - x  + 8*x/         
------------------------------
                    3/2       
      /     2      \          
      \9 - x  + 8*x/

$$- \frac{3 \left(x - 4\right) \left(\frac{\left(x - 4\right)^{2}}{- x^{2} + 8 x + 9} + 1\right)}{\left(- x^{2} + 8 x + 9\right)^{\frac{3}{2}}}$$

Simplify

The graph

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Derivative of sqrt(9+8x-x^2)

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The solution