1. The problem is to find the derivative of the function $g(x) = -6x - x^3$ and understand why the derivative is $g'(x) = -6 - 3x^2$. 2. The formula for the derivative of a function $f(x)$ is given by: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. Important rules for derivatives used here are: - The derivative of $ax^n$ is $a n x^{n-1}$ where $a$ is a constant and $n$ is a positive integer. - The derivative of a sum is the sum of the derivatives. 4. Applying the power rule to each term: - For $-6x$, the derivative is $-6 \times 1 \times x^{1-1} = -6$. - For $-x^3$, the derivative is $-1 \times 3 \times x^{3-1} = -3x^2$. 5. Adding these results gives: $$g'(x) = -6 - 3x^2$$ 6. This matches the given derivative, explaining why the answer is $-6 - 3x^2$. This process shows how to differentiate each term separately and then combine the results to get the derivative of the whole function.