Derivative of a^x*e^x

1. Apply the product rule:

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

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

   1. $\frac{\partial}{\partial x} a^{x} = a^{x} \log{\left(a \right)}$

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

   1. The derivative of $e^{x}$ is itself.

   The result is: $a^{x} e^{x} \log{\left(a \right)} + a^{x} e^{x}$

2. Now simplify:

   $$\left(e a\right)^{x} \left(\log{\left(a \right)} + 1\right)$$

The answer is:

$$\left(e a\right)^{x} \left(\log{\left(a \right)} + 1\right)$$