Derivative of t*cos(t)-2*sin(t)

1. Differentiate $t \cos{\left(t \right)} - 2 \sin{\left(t \right)}$ term by term:

   1. Apply the product rule:

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

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

      1. Apply the power rule: $t$ goes to $1$

      $g{\left(t \right)} = \cos{\left(t \right)}$; 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: $- t \sin{\left(t \right)} + \cos{\left(t \right)}$

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

      1. The derivative of sine is cosine:

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

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

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

The answer is:

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