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Derivative of:
Derivative of (x+x^2)^x
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Derivative of f(x)=1/3x²+x²+2x
Identical expressions
((sqrt2x- four)-x)^ four
(( square root of 2x minus 4) minus x) to the power of 4
(( square root of 2x minus four) minus x) to the power of four
((√2x-4)-x)^4
((sqrt2x-4)-x)4
sqrt2x-4-x4
((sqrt2x-4)-x)⁴
sqrt2x-4-x^4
Similar expressions
((sqrt2x-4)+x)^4
((sqrt2x+4)-x)^4

The derivative of the function/ ((sqrt2x-4)-x)^4

Derivative of ((sqrt2x-4)-x)^4

The solution

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                 4
/  _____        \ 
\\/ 2*x  - 4 - x/

$$\left(- x + \left(\sqrt{2 x} - 4\right)\right)^{4}$$

(sqrt(2*x) - 4 - x)^4

Detail solution

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

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

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

The answer is:

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

The graph

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The first derivative [src]

                 3 /         ___   ___\
/  _____        \  |     2*\/ 2 *\/ x |
\\/ 2*x  - 4 - x/ *|-4 + -------------|
                   \           x      /

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

Simplify

The second derivative [src]

                       /             2                              \
                     2 |  /      ___\      ___ /          ___   ___\|
/          ___   ___\  |  |    \/ 2 |    \/ 2 *\4 + x - \/ 2 *\/ x /|
\4 + x - \/ 2 *\/ x / *|3*|2 - -----|  + ---------------------------|
                       |  |      ___|                 3/2           |
                       \  \    \/ x /                x              /

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ((sqrt2x-4)-x)^4

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The solution