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Identical expressions
sqrt(twenty-eight -3x-x^ two)
square root of (28 minus 3x minus x squared )
square root of (twenty minus eight minus 3x minus x to the power of two)
√(28-3x-x^2)
sqrt(28-3x-x2)
sqrt28-3x-x2
sqrt(28-3x-x²)
sqrt(28-3x-x to the power of 2)
sqrt28-3x-x^2
Similar expressions
sqrt(28-3x+x^2)
sqrt(28+3x-x^2)

The derivative of the function/ sqrt(28-3x-x^2)

Derivative of sqrt(28-3x-x^2)

The solution

You have entered [src]

   _______________
  /             2 
\/  28 - 3*x - x

$$\sqrt{- x^{2} + \left(28 - 3 x\right)}$$

sqrt(28 - 3*x - x^2)

Detail solution

Let $u = - x^{2} + \left(28 - 3 x\right)$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x^{2} + \left(28 - 3 x\right)\right)$ :
1. Differentiate $- x^{2} + \left(28 - 3 x\right)$ term by term:
  1. Differentiate $28 - 3 x$ term by term:
    1. The derivative of the constant $28$ 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: $-3$
    The result is: $-3$
  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: $- 2 x - 3$
The result of the chain rule is:

$\frac{- 2 x - 3}{2 \sqrt{- x^{2} + \left(28 - 3 x\right)}}$
Now simplify:

$- \frac{x + \frac{3}{2}}{\sqrt{- x^{2} - 3 x + 28}}$

The answer is:

$- \frac{x + \frac{3}{2}}{\sqrt{- x^{2} - 3 x + 28}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     -3/2 - x     
------------------
   _______________
  /             2 
\/  28 - 3*x - x

$$\frac{- x - \frac{3}{2}}{\sqrt{- x^{2} + \left(28 - 3 x\right)}}$$

Simplify

The second derivative [src]

 /                 2   \ 
 |        (3 + 2*x)    | 
-|1 + -----------------| 
 |      /      2      \| 
 \    4*\28 - x  - 3*x// 
-------------------------
       _______________   
      /       2          
    \/  28 - x  - 3*x

$$- \frac{\frac{\left(2 x + 3\right)^{2}}{4 \left(- x^{2} - 3 x + 28\right)} + 1}{\sqrt{- x^{2} - 3 x + 28}}$$

Simplify

The third derivative [src]

             /               2 \
             |      (3 + 2*x)  |
-3*(3 + 2*x)*|4 + -------------|
             |          2      |
             \    28 - x  - 3*x/
--------------------------------
                       3/2      
        /      2      \         
      8*\28 - x  - 3*x/

$$- \frac{3 \left(2 x + 3\right) \left(\frac{\left(2 x + 3\right)^{2}}{- x^{2} - 3 x + 28} + 4\right)}{8 \left(- x^{2} - 3 x + 28\right)^{\frac{3}{2}}}$$

Simplify

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Derivative of sqrt(28-3x-x^2)

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The solution