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Identical expressions
(sqrt(two hundred and fifty-six -x^ two)+ two *x)/ four
( square root of (256 minus x squared ) plus 2 multiply by x) divide by 4
( square root of (two hundred and fifty minus six minus x to the power of two) plus two multiply by x) divide by four
(√(256-x^2)+2*x)/4
(sqrt(256-x2)+2*x)/4
sqrt256-x2+2*x/4
(sqrt(256-x²)+2*x)/4
(sqrt(256-x to the power of 2)+2*x)/4
(sqrt(256-x^2)+2x)/4
(sqrt(256-x2)+2x)/4
sqrt256-x2+2x/4
sqrt256-x^2+2x/4
(sqrt(256-x^2)+2*x) divide by 4
Similar expressions
(sqrt(256+x^2)+2*x)/4
(sqrt(256-x^2)-2*x)/4

The derivative of the function/ (sqrt(256-x^2)+2*x)/4

Derivative of (sqrt(256-x^2)+2*x)/4

The solution

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   __________      
  /        2       
\/  256 - x   + 2*x
-------------------
         4

$$\frac{2 x + \sqrt{- x^{2} + 256}}{4}$$

  /   __________      \
  |  /        2       |
d |\/  256 - x   + 2*x|
--|-------------------|
dx\         4         /

$$\frac{d}{d x} \frac{2 x + \sqrt{- x^{2} + 256}}{4}$$

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Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Differentiate $2 x + \sqrt{256 - x^{2}}$ term by term:
  1. Let $u = 256 - x^{2}$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(256 - x^{2}\right)$ :
    1. Differentiate $256 - x^{2}$ term by term:
      1. The derivative of the constant $256$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $- 2 x$
      The result is: $- 2 x$
    The result of the chain rule is:
    
    $- \frac{x}{\sqrt{256 - x^{2}}}$
  4. 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: $- \frac{x}{\sqrt{256 - x^{2}}} + 2$
So, the result is: $- \frac{x}{4 \sqrt{256 - x^{2}}} + \frac{1}{2}$

The answer is:

$- \frac{x}{4 \sqrt{256 - x^{2}}} + \frac{1}{2}$

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The first derivative [src]

1          x       
- - ---------------
2        __________
        /        2 
    4*\/  256 - x

$$- \frac{x}{4 \sqrt{- x^{2} + 256}} + \frac{1}{2}$$

Simplify

The second derivative [src]

           2   
          x    
 -1 + ---------
              2
      -256 + x 
---------------
     __________
    /        2 
4*\/  256 - x

$$\frac{\frac{x^{2}}{x^{2} - 256} - 1}{4 \sqrt{- x^{2} + 256}}$$

Simplify

The third derivative [src]

    /          2   \
    |         x    |
3*x*|-1 + ---------|
    |             2|
    \     -256 + x /
--------------------
              3/2   
    /       2\      
  4*\256 - x /

$$\frac{3 x \left(\frac{x^{2}}{x^{2} - 256} - 1\right)}{4 \left(- x^{2} + 256\right)^{\frac{3}{2}}}$$

Simplify

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Derivative of (sqrt(256-x^2)+2*x)/4

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The solution