Derivative of exp(a*x/2)

1. Let $u = \frac{a x}{2}$.

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

3. Then, apply the chain rule. Multiply by $\frac{\partial}{\partial x} \frac{a x}{2}$:

   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: $a$

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

   The result of the chain rule is:

   $$\frac{a e^{\frac{a x}{2}}}{2}$$

4. Now simplify:

   $$\frac{a e^{\frac{a x}{2}}}{2}$$

The answer is:

$$\frac{a e^{\frac{a x}{2}}}{2}$$