Taylor's Theorem/One Variable with Two Functions

Theorem
Let $f$ and $g$ be real functions satisfying following conditions:


 * $(1): \quad f$ is $n + 1$ times differentiable on the open interval $\openint a x$
 * $(2): \quad f$ is of differentiability class $C^n$ on the closed interval $\closedint a x$
 * $(3): \quad g$ is $k + 1$ times differentiable on the open interval $\openint a x$
 * $(4): \quad g$ is of differentiability class $C^k$ on the closed interval $\closedint a x$
 * $(5): \quad \map {g^{\paren {k + 1}}} t \ne 0$ for any $t \in \openint a x$

Then the following equation holds for some real number $\xi \in \openint a x$:
 * $\dfrac {\map {f^{\paren {n + 1} } } \xi /n!} {\map {g^{\paren {k + 1} } } \xi /k!} \paren {x - \xi}^{n - k} = \dfrac {\map f x - \map f a - \map {f'} a \paren {x - a} - \dfrac {\map {f} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {f^{\paren n} } a} {n!} \paren {x - a}^n} {\map g x - \map g a - \map {g'} a \paren {x - a} - \dfrac {\map {g} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {g^{\paren k} } a} {k!} \paren {x - a}^k}$

or equivalently:

Proof
We define $F$ and $G$ as follows:

Then $F$ and $G$ are continuous on $\closedint a x$ and differentiable on $\openint a x$.

Differentiating $t$:

By condition $(5)$ of the statement of the theorem, it follows that $G$ does not vanish on $\openint a x$.

By Cauchy Mean Value Theorem, there exists a real number $\xi \in \openint a x$ such that:
 * $\dfrac {\map {F'} \xi} {\map {G'} \xi} = \dfrac {\map F x - \map F a} {\map G x - \map G a}$

That is:
 * $\dfrac {\map {f^{\paren {n + 1} } } \xi / n!} {\map {g^{\paren {k + 1} } } \xi /k!} \paren {x - \xi}^{n - k} = \dfrac {\map f x - \map f a - \map {f'} a \paren {x - a} - \dfrac {\map {f} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {f^{\paren n} } a} {n!} \paren {x - a}^n} {\map g x - \map g a - \map {g'} a \paren {x - a} - \dfrac {\map {g} a} {2!} \paren {x - a}^2 - \dotsb - \dfrac {\map {g^{\paren k}} a} {k!} \paren {x - a}^k}$

Also see

 * Definition:Lagrange Form of Remainder of Taylor Series: taking $k = n$ and $\map g x = \paren {x - a}^{n + 1}$.
 * Definition:Cauchy Form of Remainder of Taylor Series: taking $k = 0$ and $\map g x = x - a$.