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

1. Let $u = - x + \sqrt{2 x - 3}$.

2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

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.

               1. 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:

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

4. 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: