Derivative of x^7*e^x

1. Apply the product rule:

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

   $f{\left(x \right)} = x^{7}$; to find $\frac{d}{d x} f{\left(x \right)}$:

   1. Apply the power rule: $x^{7}$ goes to $7 x^{6}$

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

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

   The result is: $x^{7} e^{x} + 7 x^{6} e^{x}$

2. Now simplify:

   $$x^{6} \left(x + 7\right) e^{x}$$

The answer is:

$$x^{6} \left(x + 7\right) e^{x}$$