Derivative of Constant Multiple/Real
Jump to navigation
Jump to search
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