Derivative of (3-4*sin(5x))^2

1. Let $u = 3 - 4 \sin{\left(5 x \right)}$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(3 - 4 \sin{\left(5 x \right)}\right)$:

   1. Differentiate $3 - 4 \sin{\left(5 x \right)}$ term by term:

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

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

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

            1. Let $u = 5 x$.

            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 $\frac{d}{d x} 5 x$:

               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: $5$

               The result of the chain rule is:

               $$5 \cos{\left(5 x \right)}$$

            So, the result is: $20 \cos{\left(5 x \right)}$

         So, the result is: $- 20 \cos{\left(5 x \right)}$

      The result is: $- 20 \cos{\left(5 x \right)}$

   The result of the chain rule is:

   $$- 20 \cdot \left(6 - 2 \cdot 4 \sin{\left(5 x \right)}\right) \cos{\left(5 x \right)}$$

4. Now simplify:

   $$40 \cdot \left(4 \sin{\left(5 x \right)} - 3\right) \cos{\left(5 x \right)}$$

The answer is:

$$40 \cdot \left(4 \sin{\left(5 x \right)} - 3\right) \cos{\left(5 x \right)}$$