Derivative of 5/(2x-3)⁴

1. The derivative of a constant times a function is the constant times the derivative of the function.

   1. Let $u = \left(2 x - 3\right)^{4}$.

   2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 3\right)^{4}$:

      1. Let $u = 2 x - 3$.

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 x - 3\right)$:

         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:

         $$8 \left(2 x - 3\right)^{3}$$

      The result of the chain rule is:

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

   So, the result is: $- \frac{40}{\left(2 x - 3\right)^{5}}$

2. Now simplify:

   $$- \frac{40}{\left(2 x - 3\right)^{5}}$$

The answer is:

$$- \frac{40}{\left(2 x - 3\right)^{5}}$$