# Power Rule for Derivatives/Corollary

Jump to navigation
Jump to search

## Corollary to Power Rule for Derivatives

Let $n \in \R$.

Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.

Then:

- $\map {\dfrac \d {\d x} } {c x^n} = n c x^{n - 1}$

everywhere that $\map f x = x^n$ is defined.

## Proof

\(\displaystyle \map {\frac \d {\d x} } {c x^n}\) | \(=\) | \(\displaystyle c \, \map {\frac \d {\d x} } {x^n}\) | Derivative of Constant Multiple | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle n c x^{n - 1}\) | Power Rule for Derivatives |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 13$: General Rules of Differentiation: $13.4$