Derivative of tg^4x+2x

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

   1. Let $u = \tan{\left(x \right)}$.

   2. Apply the power rule: $u^{4}$ goes to $4 u^{3}$

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$:

      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:

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

      The result of the chain rule is:

      $$\frac{4 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$$

   4. 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: $2$

   The result is: $\frac{4 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan^{3}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 2$

2. Now simplify:

   $$\frac{4 \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + 2$$

The answer is:

$$\frac{4 \sin^{3}{\left(x \right)}}{\cos^{5}{\left(x \right)}} + 2$$