I know there's (kind of?) some controversy as to what $g^2(x)$ means — it could be $g(g(x))$, or $(g(x))^2$ or even $g''(x)$. I'm taking a calculus course and assumed that $g^2(x)$ meant $g''(x)$, but it turns out that the solution saw this as $(g(x))^2$.

Here's the problem if it will clarify things:

If $P(x) = g^2(x)$, then $P'(x)$ equals...

(of course, there's a table given with values of g and g' accompanying the problem and answer choices, but I'm not including them here because of copyright.)

So in general, is there any way I can determine which one exactly the problem means when $g^2(x)$ appears?

  • 1
    $\begingroup$ If the problem is referring to the $n$th derivative of a function, usually it is denoted as $g^{(n)}(x)$. $\endgroup$ – JBL Apr 24 '18 at 17:15
  • 1
    $\begingroup$ Usually when I refer to the $n$th derivative with a superscript, I enclose it in parentheses. I.e., $g^{\prime\prime}(x) = g^{(2)}(x)$. $\endgroup$ – Clarinetist Apr 24 '18 at 17:15
  • $\begingroup$ Well, in my opinion that's based on context. If the end of that sentence is $g^3(x)$, then I guess you know what is being spoken about. Generally, in my experience, for the derivative, we usually use a bracket around 2 for derivatives i.e. $g''(x) = g^{(2)}(x)$ $\endgroup$ – Naweed G. Seldon Apr 24 '18 at 17:16
  • $\begingroup$ I think it's a good bet that $g^2(x)$ will always mean $g(x)^2$ in your calculus course. I have never seen it used to mean the second derivative, as others have already noted. $\endgroup$ – saulspatz Apr 24 '18 at 17:16
  • $\begingroup$ ...$2g(x)g'(x)$ $\endgroup$ – Rudi_Birnbaum Apr 24 '18 at 17:24

If it means $g''(x) $, you should write $$g^{(2)}(x) $$

If it is $g(x).g (x) $, you will write $$(g(x))^2$$

so $g^2 (x) $ means $$g (g (x)) $$

  • $\begingroup$ However, with functions $g^2(x)$ often also means $(g(x))^2$. Think of $\sin$. $\sin^2(x)$ almost always means $(\sin(x))^2$. $\endgroup$ – johnnyb Apr 24 '18 at 21:06

It usually depends on the circumstance the problem is situated in.

For example, if you're doing calculus, it's probably good notation to avoid using $g^2$ to denote the 2nd derivative and simply use $g''$ as this avoids confusion. If you're doing something like linear algebra, the notation $T^n$ would probably mean matrix multiplication, i.e., $T^n = T\circ T\circ \cdots \circ T$.

For me, I usually use the following convention:

  • $g(x)^2$ to denote square of $g(x)$
  • $g^2(x)$ to denote $g\circ g$
  • $g^{(2)}(x)$ to denote the 2nd derivative

Of course, there are probably special cases in which I may abuse notation just for sake of simplicity (e.g $\cos^2(x)$ is the square of $\cos$), but I think it's good practice to keep your notation uniform throughout your notes and textbooks. I would believe that most authors abide by this rule and would at least mention it in their text if they're changing notation.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.