Derivative of cos(5x)*e^(x/2)

1. Apply the product rule:

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

   $f{\left(x \right)} = \cos{\left(5 x \right)}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Let $u = 5 x$.

   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 $\frac{d}{d x} 5 x$:

      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: $5$

      The result of the chain rule is:

      $$- 5 \sin{\left(5 x \right)}$$

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

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

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

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

      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: $\frac{1}{2}$

      The result of the chain rule is:

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

   The result is: $- 5 e^{\frac{x}{2}} \sin{\left(5 x \right)} + \frac{e^{\frac{x}{2}} \cos{\left(5 x \right)}}{2}$

2. Now simplify:

   $$\frac{\left(- 10 \sin{\left(5 x \right)} + \cos{\left(5 x \right)}\right) e^{\frac{x}{2}}}{2}$$

The answer is:

$$\frac{\left(- 10 \sin{\left(5 x \right)} + \cos{\left(5 x \right)}\right) e^{\frac{x}{2}}}{2}$$