What does it mean when $u$ and $v$ are functions of a single variable?

Question

I am currently trying to learn how to do integration by parts again (haven't done it for years) and am stuck on the very first line of the lecture. The lecturer says that we should "recall the product rule, if u and v are functions of a single variable, then

$$(uv)' = u'v + uv'.$$

The problem is I have no idea what this means. What is the variable here they are talking about and where are the functions? If it was $y = f(x)$ I could follow it, the variable is $x$ and and $f(x)$ is the function of $y$. But the above format is foreign to me as I don't know the variable and can't see the $f()$ anywhere.

(If there is a better forum for asking these basic questions, please let me know.)

Answer 1

The $u(x)$ and $v(x)$ (or other variable instead of $x$) is implicit here. As for them both being functions of a single variable, this is necessary so the differentiation symbol of $'$, e.g., $(uv)'$, is unambiguous in terms of what the functions are being differentiated wrt to (e.g., if it was $u(x,y)$ instead, then it's not clear if you're differentiating wrt $x$ or $y$).

Answer 2

A function of a single variable is of the form:

$$ f(x)= \text{some expression involving $x$} \tag1 $$

here the variable is $x$. Sometimes one only calls this $f$ for abbreviety and when it is obvious from the text that the functions involved are single or multivariate. When one mixes in $y$ it usually because one wants to stress that $f(x)$ is also a number and $(1)$ is a relation between two numbers $x$ adn $y$.

In your question $u$ and $y$ are functions of one variable so you can think of them as $u(x)$ and $v(x)$.

Hope this helped

Answer 3

here, what they mean by a function of a single variable is a function $$ u: \Bbb R \rightarrow \Bbb R: x \mapsto u(x). $$ It is common to write $u$ instead of $u(x)$. So the product rule could be written as $$ (u\cdot v)^\prime(x) = u^\prime(x)\cdot v(x) + v^\prime(x)\cdot u(x). $$

Answer 4

How about $t$? You could think of $u$ and $v$ as functions of $t$. So you get $u(t)$ and $v(t)$.

You should remember that any letter at all could be substituted for $t$.

There's no real need to write the variable, sometimes, when it's understood. We have $(uv)'=u'v+uv'$, which is the product rule. Then we are going to integrate with respect to the variable (whatever it is).

We get $\int (uv)'=\int u'v+\int uv'$. And the left hand side is just $uv$, by the fundamental theorem of calculus.