Derivative of (5x+2)^-3

1. Let $u = 5 x + 2$.

2. Apply the power rule: $\frac{1}{u^{3}}$ goes to $- \frac{3}{u^{4}}$

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(5 x + 2\right)$:

   1. Differentiate $5 x + 2$ 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: $5$

      2. The derivative of the constant $2$ is zero.

      The result is: $5$

   The result of the chain rule is:

   $$- \frac{15}{\left(5 x + 2\right)^{4}}$$

4. Now simplify:

   $$- \frac{15}{\left(5 x + 2\right)^{4}}$$

The answer is:

$$- \frac{15}{\left(5 x + 2\right)^{4}}$$