Derivative of a sum of power functions

Rewrite $g(x)$ using negative exponents: $g(x) = x^{-1} + 3x^{-2} - 5x.$ This form makes each term ready for the power rule, since dividing by a power of $x$ is the same as multiplying by a negative power of $x$.

Differentiate each term separately using the sum/difference rule, applying the power rule $\frac{d}{dx}[x^n] = nx^{n-1}$ to each piece and the constant multiple rule where needed: $\frac{d}{dx}[x^{-1}] = -x^{-2}, \quad \frac{d}{dx}[3x^{-2}] = -6x^{-3}, \quad \frac{d}{dx}[-5x] = -5.$ Adding these results gives $g'(x) = -x^{-2} - 6x^{-3} - 5$, which can be rewritten as $g'(x) = -\frac{1}{x^2} - \frac{6}{x^3} - 5.$