Derivative of (sin(4x)+3)^3

1. Let $u = \sin{\left(4 x \right)} + 3$.

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

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

   1. Differentiate $\sin{\left(4 x \right)} + 3$ term by term:

      1. Let $u = 4 x$.

      2. The derivative of sine is cosine:

         $$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$$

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 4 x$:

         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: $4$

         The result of the chain rule is:

         $$4 \cos{\left(4 x \right)}$$

      4. The derivative of the constant $3$ is zero.

      The result is: $4 \cos{\left(4 x \right)}$

   The result of the chain rule is:

   $$12 \left(\sin{\left(4 x \right)} + 3\right)^{2} \cos{\left(4 x \right)}$$

4. Now simplify:

   $$12 \left(\sin{\left(4 x \right)} + 3\right)^{2} \cos{\left(4 x \right)}$$

The answer is:

$$12 \left(\sin{\left(4 x \right)} + 3\right)^{2} \cos{\left(4 x \right)}$$