Derivative of (1+tan(3*x)^(2))*e^(-x/2)

1. Apply the product rule:

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

   $f{\left(x \right)} = \tan^{2}{\left(3 x \right)} + 1$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Differentiate $\tan^{2}{\left(3 x \right)} + 1$ term by term:

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

      2. Let $u = \tan{\left(3 x \right)}$.

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

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

         1. Rewrite the function to be differentiated:

            $$\tan{\left(3 x \right)} = \frac{\sin{\left(3 x \right)}}{\cos{\left(3 x \right)}}$$

         2. Apply the quotient rule, which is:

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

            $f{\left(x \right)} = \sin{\left(3 x \right)}$ and $g{\left(x \right)} = \cos{\left(3 x \right)}$.

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

            1. Let $u = 3 x$.

            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 $\frac{d}{d x} 3 x$:

               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:

               $$3 \cos{\left(3 x \right)}$$

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

            1. Let $u = 3 x$.

            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 $\frac{d}{d x} 3 x$:

               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:

               $$- 3 \sin{\left(3 x \right)}$$

            Now plug in to the quotient rule:

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

         The result of the chain rule is:

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

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

   $g{\left(x \right)} = e^{\frac{\left(-1\right) x}{2}}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. Let $u = \frac{\left(-1\right) x}{2}$.

   2. The derivative of $e^{u}$ is itself.

   3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\left(-1\right) x}{2}$:

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

         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: $-1$

         So, the result is: $- \frac{1}{2}$

      The result of the chain rule is:

      $$- \frac{e^{\frac{\left(-1\right) x}{2}}}{2}$$

   The result is: $\frac{2 \left(3 \sin^{2}{\left(3 x \right)} + 3 \cos^{2}{\left(3 x \right)}\right) e^{\frac{\left(-1\right) x}{2}} \tan{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} - \frac{\left(\tan^{2}{\left(3 x \right)} + 1\right) e^{\frac{\left(-1\right) x}{2}}}{2}$

2. Now simplify:

   $$\frac{\left(12 \tan{\left(3 x \right)} - 1\right) e^{- \frac{x}{2}}}{2 \cos^{2}{\left(3 x \right)}}$$

The answer is:

$$\frac{\left(12 \tan{\left(3 x \right)} - 1\right) e^{- \frac{x}{2}}}{2 \cos^{2}{\left(3 x \right)}}$$