Derivative of а(sint-tcost)

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

   1. Differentiate $- tcost + \sin{\left(t \right)}$ term by term:

      1. The derivative of sine is cosine:

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

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

      The result is: $\cos{\left(t \right)}$

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

The answer is:

$$a \cos{\left(t \right)}$$