4.1 Power, Constant Multiple and Sum Rules

Notes

These rules handle most polynomial derivatives.

• Constant: $\displaystyle \frac{d}{dx}[c] = 0$.
• Power rule: $\displaystyle \frac{d}{dx}[x^n] = nx^{n-1}$ for any real $n$.
• Constant multiple: $\displaystyle \frac{d}{dx}[c\,f(x)] = c\,f'(x)$.
• Sum/difference: differentiate term by term.
• Example: $\displaystyle \frac{d}{dx}[4x^3 - 6x + 7] = 12x^2 - 6$. Rewrite roots and fractions as powers first: $\displaystyle \frac{1}{x^2} = x^{-2}$.