Find the derivative of the function $g(x) = x^2 - 8\ln{x} - 1$.

1. Problem Description

Find the derivative of the function $g(x) = x^2 - 8\ln{x} - 1$.

2. Solution Steps

To find the derivative of $g(x) = x^2 - 8\ln{x} - 1$, we need to differentiate each term with respect to $x$.

The derivative of $x^2$ is $2x$.

The derivative of $\ln{x}$ is $\frac{1}{x}$. Therefore, the derivative of $8\ln{x}$ is $8 \cdot \frac{1}{x} = \frac{8}{x}$.

The derivative of a constant is $0$. So, the derivative of $-1$ is $0$.

Therefore, the derivative of $g(x)$ is given by:

$g'(x) = \frac{d}{dx}(x^2) - \frac{d}{dx}(8\ln{x}) - \frac{d}{dx}(1)$

$g'(x) = 2x - \frac{8}{x} - 0$

$g'(x) = 2x - \frac{8}{x}$

3. Final Answer

The derivative of $g(x)$ is $g'(x) = 2x - \frac{8}{x}$.