Derivative of 4/(9+7x)^5

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

   1. Let $u = \left(7 x + 9\right)^{5}$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(7 x + 9\right)^{5}$:

      1. Let $u = 7 x + 9$.

      2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(7 x + 9\right)$:

         1. Differentiate $7 x + 9$ term by term:

            1. The derivative of the constant $9$ is zero.

            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: $7$

            The result is: $7$

         The result of the chain rule is:

         $$35 \left(7 x + 9\right)^{4}$$

      The result of the chain rule is:

      $$- \frac{35}{\left(7 x + 9\right)^{6}}$$

   So, the result is: $- \frac{140}{\left(7 x + 9\right)^{6}}$

The answer is:

$$- \frac{140}{\left(7 x + 9\right)^{6}}$$