Derivative of a sum using linearity rules

Apply the sum/difference rule to break the derivative into three pieces: $g'(x) = \frac{d}{dx}[5\sin(x)] - \frac{d}{dx}[3x^2] + \frac{d}{dx}[8]$. Each term can now be differentiated independently.

Using the constant multiple rule, $\frac{d}{dx}[5\sin(x)] = 5\cos(x)$ and $\frac{d}{dx}[3x^2] = 3(2x) = 6x$. Since $8$ is a constant, its derivative is $0$ by the constant rule. Combining these gives $g'(x) = 5\cos(x) - 6x$.

The constant term, eight, becomes zero after differentiating. By the constant rule, the derivative of any constant is zero, so it disappears entirely from g prime, leaving only the terms coming from five sine x and negative three x squared.