Derivative of (5x-3)(2x^3+3)

1. Apply the product rule:

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

   $f{\left(x \right)} = 5 x - 3$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $5 x - 3$ 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: $5$

      2. The derivative of the constant $-3$ is zero.

      The result is: $5$

   $g{\left(x \right)} = 2 x^{3} + 3$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Differentiate $2 x^{3} + 3$ 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: $x^{3}$ goes to $3 x^{2}$

         So, the result is: $6 x^{2}$

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

      The result is: $6 x^{2}$

   The result is: $10 x^{3} + 6 x^{2} \left(5 x - 3\right) + 15$

2. Now simplify:

   $$40 x^{3} - 18 x^{2} + 15$$

The answer is:

$$40 x^{3} - 18 x^{2} + 15$$