The Notation for Derivatives

Question

"The derivative of a sum is the sum of derivatives"

Above theorem can be mathematically expressed as:

$$h'(x)=f'(x)+g'(x)$$

where $f(x)$ and $g(x)$ are two differentiable functions. What is the right way to express the statement of this theorem in Leibniz notation? Is it $$\frac{d}{dx}h=\frac{d}{dx}(f+g)=\frac{d}{dx}f+\frac{d}{dx}g$$

OR

$$\frac{d}{dx}h(x)=\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)?$$

In other words, is it permissible to write the derivative of functions in such a way so as to express the variable(s) on which they depend?

Answer 1

Honestly, you should also write $$h'=f'+g',$$ since you take derivative of functions, and not of values of functions. Anyway, Leibniz' notation emphasizes the variable (you can't write $\frac{d}{dx} \sin t = \cos t$), so you might prefer to write $\frac{d}{dx}h(x) = \frac{d}{dx} f(x) + \frac{d}{dx}g(x)$. My favorite notation remains $Dh=Df+Dg$, which is rather common in infinite-dimensional settings: it stresses the functional nature of differentiation.

Answer 2

it's common to write the variable to express that it's the variable you are deriving according to. but anyway it's not that critical

Answer 3

I have only seen $\frac{d}{dx}h(x)=\frac{d}{dx}(f(x)+g(x))=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)$.

Even with the prime notation, you don't say $h'=f'+g'$, unless it is for shorthand for scrap work. If it is being graded, I would always specify that the function is some function of a variable- in this case $x$.

Answer 4

$f(x)$ is a value, not a function, so technically it's wrong to say that $f(x)$ is differentiable. To be precise you should say "$h'(x) = f'(x) + g'(x)$ for any $x \in D$ for any differentiable functions $f,g$ where $h(x) = f(x) + g(x)$ for any $x \in D$", but it is more concise and equally precise to say "$(f+g)' = f' + g'$ on D". Likewise the middle line with $\frac{d}{dx} f$ is probably the 'original' Leibniz notation, but it makes no sense if $f,g,h$ are strictly functions; the only way to get the correct semantic meaning is to interpret $f,g,h$ not as functions but as values that are determined by $x$. Historically $\frac{df}{dx}$ meant the change in $f$ divided by the change in $x$, with "d" standing for "Δ" / "δ" (delta). But if you want to interpret $f,g,h$ as true (1-variable) functions, then none of them are externally bound to any variables, so $x$ has no relation to $f,g,h$.

At the same time, Wikipedia says that Leibniz's notation is indeed in the form of the last line, but again we cannot interpret $f(x)$ within the $\frac{d}{dx}$ as the value of $f$ on input $x$, but rather have to interpret $\frac{d}{dx}(f(x))$ as something more like $\frac{d}{dx}($"f(x)"$)$ where $\frac{d}{dx}$ behaves like a procedure that looks at how the value described by the input string containing some occurrences of "$x$" depends on the value of $x$. This somewhat strange abuse of notation is quite common in many areas of mathematics, which you should be aware of, and when using such notation you should be clear about which expressions denote the objects themselves, which denote their descriptions, and which denote their state as bound to the state of other objects.

Answer 5

There's a distinction between function and expression.

If $f$ and $g$ are differentiable functions $f,g : I \rightarrow \mathbb{R}$ where $I$ is an open set, then the correct notation is:

$$\frac{d}{dx}(f(x)+g(x)) = \frac{d}{dx}f(x) + \frac{d}{dx}g(x)$$

If, on the other hand, $f$ and $g$ are expressions of type $\mathbb{R}$ having a free variable $x$ bound to the set $I$, and if $f$ and $g$ are furthermore differentiable in $x$, then the correct notation is:

$$\frac{d}{dx}(f+g) = \frac{d}{dx}f + \frac{d}{dx}g$$