Derivative of sqrt(x^1/3)*e^2x

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 $1$

      $g{\left(x \right)} = \sqrt{\sqrt[3]{x}}$; to find $\frac{d}{d x} g{\left(x \right)}$:

      1. Let $u = \sqrt[3]{x}$.

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

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

         1. Apply the power rule: $\sqrt[3]{x}$ goes to $\frac{1}{3 x^{\frac{2}{3}}}$

         The result of the chain rule is:

         $$\frac{1}{6 x^{\frac{5}{6}}}$$

      The result is: $\frac{\sqrt[6]{x}}{6} + \sqrt{\sqrt[3]{x}}$

   So, the result is: $\left(\frac{\sqrt[6]{x}}{6} + \sqrt{\sqrt[3]{x}}\right) e^{2}$

2. Now simplify:

   $$\frac{7 \sqrt[6]{x} e^{2}}{6}$$

The answer is:

$$\frac{7 \sqrt[6]{x} e^{2}}{6}$$