Power Rule for Derivatives/Corollary

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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

$\blacksquare$


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