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

1. Let $u = \sin{\left(3 x^{2} \cdot 4 x + 5 \right)}$.

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

3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x^{2} \cdot 4 x + 5 \right)}$:

   1. Let $u = 3 x^{2} \cdot 4 x + 5$.

   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 $\frac{d}{d x} \left(3 x^{2} \cdot 4 x + 5\right)$:

      1. Differentiate $3 x^{2} \cdot 4 x + 5$ term by term:

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

            1. Apply the product rule:

               $$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$$

               $f{\left(x \right)} = x$; to find $\frac{d}{d x} f{\left(x \right)}$:

               1. Apply the power rule: $x$ goes to $1$

               $g{\left(x \right)} = x^{2}$; to find $\frac{d}{d x} g{\left(x \right)}$:

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

               The result is: $3 x^{2}$

            So, the result is: $36 x^{2}$

         2. The derivative of the constant $5$ is zero.

         The result is: $36 x^{2}$

      The result of the chain rule is:

      $$36 x^{2} \cos{\left(3 x^{2} \cdot 4 x + 5 \right)}$$

   The result of the chain rule is:

   $$108 x^{2} \sin^{2}{\left(3 x^{2} \cdot 4 x + 5 \right)} \cos{\left(3 x^{2} \cdot 4 x + 5 \right)}$$

4. Now simplify:

   $$108 x^{2} \sin^{2}{\left(12 x^{3} + 5 \right)} \cos{\left(12 x^{3} + 5 \right)}$$

The answer is:

$$108 x^{2} \sin^{2}{\left(12 x^{3} + 5 \right)} \cos{\left(12 x^{3} + 5 \right)}$$