Derivative of tan(21x)+tan(3x)

1. Differentiate $\tan{\left(3 x \right)} + \tan{\left(21 x \right)}$ term by term:

   1. Rewrite the function to be differentiated:

      $$\tan{\left(21 x \right)} = \frac{\sin{\left(21 x \right)}}{\cos{\left(21 x \right)}}$$

   2. Apply the quotient rule, which is:

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

      $f{\left(x \right)} = \sin{\left(21 x \right)}$ and $g{\left(x \right)} = \cos{\left(21 x \right)}$.

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

      1. Let $u = 21 x$.

      2. The derivative of sine is cosine:

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 21 x$:

         1. 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: $21$

         The result of the chain rule is:

         $$21 \cos{\left(21 x \right)}$$

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

      1. Let $u = 21 x$.

      2. The derivative of cosine is negative sine:

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

      3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 21 x$:

         1. 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: $21$

         The result of the chain rule is:

         $$- 21 \sin{\left(21 x \right)}$$

      Now plug in to the quotient rule:

      $\frac{21 \sin^{2}{\left(21 x \right)} + 21 \cos^{2}{\left(21 x \right)}}{\cos^{2}{\left(21 x \right)}}$

   3. Let $u = 3 x$.

   4. $$\frac{d}{d u} \tan{\left(u \right)} = \frac{1}{\cos^{2}{\left(u \right)}}$$

   5. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x$:

      1. 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: $3$

      The result of the chain rule is:

      $$\frac{3}{\cos^{2}{\left(3 x \right)}}$$

   The result is: $\frac{3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + \frac{21 \sin^{2}{\left(21 x \right)} + 21 \cos^{2}{\left(21 x \right)}}{\cos^{2}{\left(21 x \right)}}$

2. Now simplify:

   $$\frac{21}{\cos^{2}{\left(21 x \right)}} + \frac{3}{\cos^{2}{\left(3 x \right)}}$$

The answer is:

$$\frac{21}{\cos^{2}{\left(21 x \right)}} + \frac{3}{\cos^{2}{\left(3 x \right)}}$$