Derivative of a polynomial evaluated at a point

Apply the sum/difference rule and constant multiple rule to differentiate each term separately: $\frac{d}{dx}[3x^4] = 12x^3$, $\frac{d}{dx}[2x^3] = 6x^2$, $\frac{d}{dx}[-x] = -1$, and $\frac{d}{dx}[9] = 0$ since it is a constant. Combining these gives $g'(x) = 12x^3 + 6x^2 - 1$.

Now substitute $x = 2$ into $g'(x)$: $g'(2) = 12(2)^3 + 6(2)^2 - 1 = 12(8) + 6(4) - 1 = 96 + 24 - 1$. Adding these terms gives $g'(2) = 119$.

The constant term nine differentiates to zero, since the constant rule states that the derivative of any constant is zero. This is why the final expression for g prime of x contains no leftover number term from the original nine. Only the terms involving x contributed to g prime of x. The answer is zero.