Derivative of Constant Multiple/Real

Theorem

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

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


Then:

$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$


Corollary

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


Proof

\(\ds \map {\dfrac \d {\d x} } {c \map f x}\) \(=\) \(\ds c \map {\dfrac \d {\d x} } {\map f x} + \map f x \map {\dfrac \d {\d x} } c\) Product Rule for Derivatives
\(\ds \) \(=\) \(\ds c \map {\dfrac \d {\d x} } {\map f x} + 0\) Derivative of Constant
\(\ds \) \(=\) \(\ds c \map {\dfrac \d {\d x} } {\map f x}\)

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: General Rules of Differentiation: $13.6$
  • 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $5$. Differential Calculus: Appendix: Differentiation Rules: $1.$ Constant factor
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