Power Rule for Derivatives/Corollary

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

\(\ds \map {\frac \d {\d x} } {c x^n}\)

\(=\)

\(\ds c \, \map {\frac \d {\d x} } {x^n}\)

Derivative of Constant Multiple

\(\ds \)

\(=\)

\(\ds n c x^{n - 1}\)

Power Rule for Derivatives

$\blacksquare$

Sources

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