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Identical expressions
three *sqrt(two x)+ five *sqrt(2* six thousand, eight hundred -2x)
3 multiply by square root of (2x) plus 5 multiply by square root of (2 multiply by 6800 minus 2x)
three multiply by square root of (two x) plus five multiply by square root of (2 multiply by six thousand, eight hundred minus 2x)
3*√(2x)+5*√(2*6800-2x)
3sqrt(2x)+5sqrt(26800-2x)
3sqrt2x+5sqrt26800-2x
Similar expressions
3*sqrt(2x)-5*sqrt(2*6800-2x)
3*sqrt(2x)+5*sqrt(2*6800+2x)

The derivative of the function/ 3*sqrt(2x)+5*sqrt(2*6800-2x)

Derivative of 3sqrt(2x)+5sqrt(2*6800-2x)

The solution

You have entered [src]

    _____       ______________
3*\/ 2*x  + 5*\/ 2*6800 - 2*x

$$3 \sqrt{2 x} + 5 \sqrt{- 2 x + 2 \cdot 6800}$$

d /    _____       ______________\
--\3*\/ 2*x  + 5*\/ 2*6800 - 2*x /
dx

$$\frac{d}{d x} \left(3 \sqrt{2 x} + 5 \sqrt{- 2 x + 2 \cdot 6800}\right)$$

Transform

Detail solution

Differentiate $3 \sqrt{2 x} + 5 \sqrt{- 2 x + 2 \cdot 6800}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 2 x$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. 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 of the chain rule is:
    
    $\frac{\sqrt{2}}{2 \sqrt{x}}$
  So, the result is: $\frac{3 \sqrt{2}}{2 \sqrt{x}}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = - 2 x + 2 \cdot 6800$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- 2 x + 2 \cdot 6800\right)$ :
    1. Differentiate $- 2 x + 2 \cdot 6800$ term by term:
      1. The derivative of the constant $2 \cdot 6800$ is zero.
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
        
        So, the result is: $-2$
      The result is: $-2$
    The result of the chain rule is:
    
    $- \frac{1}{\sqrt{- 2 x + 2 \cdot 6800}}$
  So, the result is: $- \frac{5}{\sqrt{- 2 x + 2 \cdot 6800}}$
The result is: $- \frac{5}{\sqrt{- 2 x + 2 \cdot 6800}} + \frac{3 \sqrt{2}}{2 \sqrt{x}}$
Now simplify:

$- \frac{5 \sqrt{2}}{2 \sqrt{6800 - x}} + \frac{3 \sqrt{2}}{2 \sqrt{x}}$

The answer is:

$- \frac{5 \sqrt{2}}{2 \sqrt{6800 - x}} + \frac{3 \sqrt{2}}{2 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                         ___
         5           3*\/ 2 
- ---------------- + -------
    ______________       ___
  \/ 2*6800 - 2*x    2*\/ x

$$- \frac{5}{\sqrt{- 2 x + 2 \cdot 6800}} + \frac{3 \sqrt{2}}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

   ___ / 3           5      \ 
-\/ 2 *|---- + -------------| 
       | 3/2             3/2| 
       \x      (6800 - x)   / 
------------------------------
              4

$$- \frac{\sqrt{2} \cdot \left(\frac{5}{\left(6800 - x\right)^{\frac{3}{2}}} + \frac{3}{x^{\frac{3}{2}}}\right)}{4}$$

Simplify

The third derivative [src]

    ___ /        5          3  \
3*\/ 2 *|- ------------- + ----|
        |            5/2    5/2|
        \  (6800 - x)      x   /
--------------------------------
               8

$$\frac{3 \sqrt{2} \left(- \frac{5}{\left(6800 - x\right)^{\frac{5}{2}}} + \frac{3}{x^{\frac{5}{2}}}\right)}{8}$$

Simplify

The graph

Derivative of 3*sqrt(2x)+5*sqrt(2*6800-2x)

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Derivative of 3sqrt(2x)+5sqrt(2*6800-2x)

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The solution

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Derivative of 3*sqrt(2x)+5*sqrt(2*6800-2x)

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The solution

Derivative of 3sqrt(2x)+5sqrt(2*6800-2x)