Derivative of f(x)=20(7*x+4)^4

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

   1. Let $u = 7 x + 4$.

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

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(7 x + 4\right)$:

      1. Differentiate $7 x + 4$ 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: $7$

         2. The derivative of the constant $4$ is zero.

         The result is: $7$

      The result of the chain rule is:

      $$28 \left(7 x + 4\right)^{3}$$

   So, the result is: $560 \left(7 x + 4\right)^{3}$

2. Now simplify:

   $$560 \left(7 x + 4\right)^{3}$$

The answer is:

$$560 \left(7 x + 4\right)^{3}$$