Derivative of tgx-2cosx

1. Differentiate $- 2 \cos{\left(x \right)} + \tan{\left(x \right)}$ term by term:

   1. Rewrite the function to be differentiated:

      $$\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$$

   2. Apply the quotient rule, which is:

      $$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$$

      $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$.

      To find $\frac{d}{d x} f{\left(x \right)}$:

      1. The derivative of sine is cosine:

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

      To find $\frac{d}{d x} g{\left(x \right)}$:

      1. The derivative of cosine is negative sine:

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

      Now plug in to the quotient rule:

      $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$

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

      1. The derivative of cosine is negative sine:

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

      So, the result is: $2 \sin{\left(x \right)}$

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

2. Now simplify:

   $$\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)} + 2}{\cos{\left(2 x \right)} + 1}$$

The answer is:

$$\frac{\sin{\left(x \right)} + \sin{\left(3 x \right)} + 2}{\cos{\left(2 x \right)} + 1}$$