Derivative of (6-x^3)*sin5x

1. Apply the product rule:

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

   $f{\left(x \right)} = 6 - x^{3}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $6 - x^{3}$ term by term:

      1. The derivative of the constant $6$ 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: $x^{3}$ goes to $3 x^{2}$

         So, the result is: $- 3 x^{2}$

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

   $g{\left(x \right)} = \sin{\left(5 x \right)}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   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:

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

   The result is: $- 3 x^{2} \sin{\left(5 x \right)} + 5 \left(6 - x^{3}\right) \cos{\left(5 x \right)}$

2. Now simplify:

   $$- 3 x^{2} \sin{\left(5 x \right)} - \left(5 x^{3} - 30\right) \cos{\left(5 x \right)}$$

The answer is:

$$- 3 x^{2} \sin{\left(5 x \right)} - \left(5 x^{3} - 30\right) \cos{\left(5 x \right)}$$