Linear Combination of Derivatives

Theorem

Let $\map f x, \map g x$ be real functions defined on the open interval $I$.

Let $\xi \in I$ be a point in $I$ at which both $f$ and $g$ are differentiable.


Then:

$\map D {\lambda f + \mu g} = \lambda D f + \mu D g$

at the point $\xi$ for some $\lambda \in \R$ and $\mu \in \R$.


It follows from the definition of derivative that if $f$ and $g$ are both differentiable on the interval $I$, then:

$\forall x \in I: \map D {\lambda \map f x + \mu \map g x} = \lambda D \map f x + \mu D \map g x$


Proof


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Let $\xi \in I$ be a point in $I$ at which $f$ is differentiable. Then by definition, $f$ is differentiable at the point $\xi$ implies that the limit:

$\ds \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$

exists.


Let $\xi \in I$ be a point in $I$ at which $g$ is differentiable. Then by definition, $g$ is differentiable at the point $\xi$ implies that the limit:

$\ds \lim_{h \mathop \to 0} \frac {\map g {\xi + h} - \map g \xi} h$

exists.

So the limit:

$\ds \lim_{h \mathop \to 0} \lambda \paren {\frac {\map f {\xi + h} - \map f \xi} h} + \mu \paren {\frac {\map g {\xi + h} - \map g \xi} h}$

exists.

Since $f$ and $g$ are real functions that are differentiable at $\xi \in I$, and $\lambda, \mu \in \R$, it follows that:

$\map{ \paren{\lambda f + \mu g} } {x}$

exists, and is differentiable at $\xi \in I$ real function.

So the limit:

$\ds \lim_{h \mathop \to 0} \frac { \map{ \paren{\lambda f + \mu g} } {\xi + h} - \map{ \paren{\lambda f + \mu g} } \xi} h$

exists.

We have:

$\ds \lim_{h \mathop \to 0} \frac { \map{ \paren{\lambda f + \mu g} } {\xi + h} - \map{ \paren{\lambda f + \mu g} } \xi} h = \ds \lim_{h \mathop \to 0} \lambda \paren {\frac {\map f {\xi + h} - \map f \xi} h} + \mu \paren {\frac {\map g {\xi + h} - \map g \xi} h} $

Hence, the result follows from the definition of derivative.

$\blacksquare$


Sources

  • 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 10.9 \ \text{(i)}$
  • 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.14$: Second Order Linear Equations: Introduction
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