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Identical expressions
(sqrt(two *x- three)-x)^ four
( square root of (2 multiply by x minus 3) minus x) to the power of 4
( square root of (two multiply by x minus three) minus x) to the power of four
(√(2*x-3)-x)^4
(sqrt(2*x-3)-x)4
sqrt2*x-3-x4
(sqrt(2*x-3)-x)⁴
(sqrt(2x-3)-x)^4
(sqrt(2x-3)-x)4
sqrt2x-3-x4
sqrt2x-3-x^4
Similar expressions
(sqrt(2*x-3)+x)^4
(sqrt(2*x+3)-x)^4

The derivative of the function/ (sqrt(2*x-3)-x)^4

Derivative of (sqrt(2*x-3)-x)^4

The solution

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                 4
/  _________    \ 
\\/ 2*x - 3  - x/

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

(sqrt(2*x - 3) - x)^4

Detail solution

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

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                      /                                                /         1      \ /      __________\\
                      |                                          2   3*|1 - ------------|*\x - \/ -3 + 2*x /|
                      |                    3   /      __________\      |      __________|                   |
   /      __________\ |  /         1      \    \x - \/ -3 + 2*x /      \    \/ -3 + 2*x /                   |
12*\x - \/ -3 + 2*x /*|2*|1 - ------------|  - ------------------- + ---------------------------------------|
                      |  |      __________|                 5/2                             3/2             |
                      \  \    \/ -3 + 2*x /       (-3 + 2*x)                      (-3 + 2*x)                /

$$12 \left(x - \sqrt{2 x - 3}\right) \left(2 \left(1 - \frac{1}{\sqrt{2 x - 3}}\right)^{3} + \frac{3 \left(1 - \frac{1}{\sqrt{2 x - 3}}\right) \left(x - \sqrt{2 x - 3}\right)}{\left(2 x - 3\right)^{\frac{3}{2}}} - \frac{\left(x - \sqrt{2 x - 3}\right)^{2}}{\left(2 x - 3\right)^{\frac{5}{2}}}\right)$$

Simplify

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Derivative of (sqrt(2*x-3)-x)^4

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The solution