Derivative of Function plus Constant

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

\(\ds \)

\(=\)

\(\ds \map {\dfrac \d {\d x} } {\map f x}\)

$\blacksquare$