Derivative of 3(cos(t)+(t)sin(t))

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

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

      1. The derivative of cosine is negative sine:

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

      2. 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 $1$

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

         1. The derivative of sine is cosine:

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

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

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

   So, the result is: $3 t \cos{\left(t \right)}$

The answer is:

$$3 t \cos{\left(t \right)}$$