Derivative of 2*tan(x)-4*x+pi+13

1. Differentiate $\left(\left(- 4 x + 2 \tan{\left(x \right)}\right) + \pi\right) + 13$ term by term:

   1. Differentiate $\left(- 4 x + 2 \tan{\left(x \right)}\right) + \pi$ term by term:

      1. Differentiate $- 4 x + 2 \tan{\left(x \right)}$ term by term:

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

            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:

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

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

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

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

            So, the result is: $-4$

         The result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 4$

      2. The derivative of the constant $\pi$ is zero.

      The result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 4$

   2. The derivative of the constant $13$ is zero.

   The result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - 4$

2. Now simplify:

   $$2 \tan^{2}{\left(x \right)} - 2$$

The answer is:

$$2 \tan^{2}{\left(x \right)} - 2$$