Derivative of 3a*sin(t^2)*cos(t)

1. Apply the product rule:

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

   $f{\left(t \right)} = 3 a \sin{\left(t^{2} \right)}$; to find $\frac{d}{d t} f{\left(t \right)}$:

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

      1. Let $u = t^{2}$.

      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 $\frac{d}{d t} t^{2}$:

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

         The result of the chain rule is:

         $$2 t \cos{\left(t^{2} \right)}$$

      So, the result is: $6 a t \cos{\left(t^{2} \right)}$

   $g{\left(t \right)} = \cos{\left(t \right)}$; to find $\frac{d}{d t} g{\left(t \right)}$:

   1. The derivative of cosine is negative sine:

      $$\frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)}$$

   The result is: $6 a t \cos{\left(t \right)} \cos{\left(t^{2} \right)} - 3 a \sin{\left(t \right)} \sin{\left(t^{2} \right)}$

2. Now simplify:

   $$3 a \left(2 t \cos{\left(t \right)} \cos{\left(t^{2} \right)} - \sin{\left(t \right)} \sin{\left(t^{2} \right)}\right)$$

The answer is:

$$3 a \left(2 t \cos{\left(t \right)} \cos{\left(t^{2} \right)} - \sin{\left(t \right)} \sin{\left(t^{2} \right)}\right)$$