Derivative of Constant Multiple/Real/Corollary

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Corollary to Derivative of Constant Multiple

Let $f$ be a real function which is differentiable on $\R$.

Let $c \in \R$ be a constant.


Then:

$D^n_x \left({c f \left({x}\right)}\right) = c D^n_{x} \left({f \left({x}\right)}\right)$


Proof


By induction: the base case is for $n = 1$ and is proved in Derivative of Constant Multiple.

Now consider $D^{k + 1}_x \left({c f \left({x}\right)}\right)$, assuming the induction hypothesis $D^k_x \left({c f \left({x}\right)}\right) = c D^k_x \left({f \left({x}\right)}\right)$:

\(\displaystyle D^{k + 1}_x \left({c f \left({x}\right)}\right)\) \(=\) \(\displaystyle D_x \left({D^k_x \left({c f \left({x}\right)}\right)}\right)\) $\quad$ Definition of Higher Derivative $\quad$
\(\displaystyle \) \(=\) \(\displaystyle D_x \left({c D^k_x \left({f \left({x}\right)}\right)}\right)\) $\quad$ Induction Hypothesis $\quad$
\(\displaystyle \) \(=\) \(\displaystyle c D_x \left({D^k_x \left({f \left({x}\right)}\right)}\right)\) $\quad$ Base Case $\quad$
\(\displaystyle \) \(=\) \(\displaystyle c D^{k + 1}_x \left({f \left({x}\right)}\right)\) $\quad$ Definition of Higher Derivative $\quad$

Hence the result by induction.

$\blacksquare$