Derivative of 2*sin(t)*cos(t)+1

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

   1. Apply the product rule:

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

      $f{\left(t \right)} = 2 \sin{\left(t \right)}$; 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. The derivative of sine is cosine:

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

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

      $g{\left(t \right)} = \cos{\left(t \right)}$; 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: $- 2 \sin^{2}{\left(t \right)} + 2 \cos^{2}{\left(t \right)}$

   2. The derivative of the constant $1$ is zero.

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

2. Now simplify:

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

The answer is:

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