Derivative of exp(0.5*x*sqrt(x))

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

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

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

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

      1. Apply the product rule:

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

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

         1. Apply the power rule: $x$ goes to $1$

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

         1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$

         The result is: $\frac{3 \sqrt{x}}{2}$

      So, the result is: $\frac{3 \sqrt{x}}{4}$

   The result of the chain rule is:

   $$\frac{3 \sqrt{x} e^{\frac{x^{\frac{3}{2}}}{2}}}{4}$$

The answer is:

$$\frac{3 \sqrt{x} e^{\frac{x^{\frac{3}{2}}}{2}}}{4}$$