Derivative of Function plus Constant
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Theorem
Let $f$ be a real function which is differentiable on $\R$.
Let $c \in \R$ be a constant.
Then:
- $\map {\dfrac \d {\d x} } {\map f x + c} = \map {\dfrac \d {\d x} } {\map f x}$
Proof
\(\ds \map {\dfrac \d {\d x} } {\map f x + c}\) | \(=\) | \(\ds \map {\dfrac \d {\d x} } {\map f x} + \map f x \, c\) | Sum Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\dfrac \d {\d x} } {\map f x} + 0\) | Derivative of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\dfrac \d {\d x} } {\map f x}\) |
$\blacksquare$