Derivative of x*10^x

1. Apply the product rule:

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

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

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

   $g{\left(x \right)} = 10^{x}$; to find $\frac{d}{d x} g{\left(x \right)}$:

   1. $\frac{d}{d x} 10^{x} = 10^{x} \log{\left(10 \right)}$

   The result is: $10^{x} x \log{\left(10 \right)} + 10^{x}$

2. Now simplify:

   $$10^{x} \left(x \log{\left(10 \right)} + 1\right)$$

The answer is:

$$10^{x} \left(x \log{\left(10 \right)} + 1\right)$$