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Identical expressions
sqrt(three + two x-x^2)
square root of (3 plus 2x minus x squared )
square root of (three plus two x minus x squared )
√(3+2x-x^2)
sqrt(3+2x-x2)
sqrt3+2x-x2
sqrt(3+2x-x²)
sqrt(3+2x-x to the power of 2)
sqrt3+2x-x^2
Similar expressions
sqrt(3+2x+x^2)
sqrt(3-2x-x^2)

The derivative of the function/ sqrt(3+2x-x^2)

Derivative of sqrt(3+2x-x^2)

The solution

You have entered [src]

   ______________
  /            2 
\/  3 + 2*x - x

$$\sqrt{- x^{2} + \left(2 x + 3\right)}$$

sqrt(3 + 2*x - x^2)

Detail solution

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

$\frac{2 - 2 x}{2 \sqrt{- x^{2} + \left(2 x + 3\right)}}$
Now simplify:

$\frac{1 - x}{\sqrt{- x^{2} + 2 x + 3}}$

The answer is:

$\frac{1 - x}{\sqrt{- x^{2} + 2 x + 3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1 - x      
-----------------
   ______________
  /            2 
\/  3 + 2*x - x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of sqrt(3+2x-x^2)

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The solution