Derivative of f(x)=-3sin(5x-6)+12x²

1. Differentiate $12 x^{2} - 3 \sin{\left(5 x - 6 \right)}$ term by term:

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

      1. Let $u = 5 x - 6$.

      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 $\frac{d}{d x} \left(5 x - 6\right)$:

         1. Differentiate $5 x - 6$ 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 $\left(-1\right) 6$ is zero.

            The result is: $5$

         The result of the chain rule is:

         $$5 \cos{\left(5 x - 6 \right)}$$

      So, the result is: $- 15 \cos{\left(5 x - 6 \right)}$

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

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

      So, the result is: $24 x$

   The result is: $24 x - 15 \cos{\left(5 x - 6 \right)}$

2. Now simplify:

   $$24 x - 15 \cos{\left(5 x - 6 \right)}$$

The answer is:

$$24 x - 15 \cos{\left(5 x - 6 \right)}$$