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  1. $\sin^2 x $

  2. $(\sin x)^2 $

Yes, I know that these two are interchangeable.

But why must we change to the second form before we differentiate it?

What's the explanation behind this theory?

E. Joseph
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user307640
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    You don't have to but it allows you to see how the power rule applies. – Sean Roberson May 27 '17 at 14:53
  • People often jump too conclusion that the main function of $\sin^2x$ is $\sin x$ so for example they derive it like $\cos^2 x\cdot (\cos x)$ or something like that (I've witnessed something similar happen with $\sin^2 x$).Instead of deriving it like $((\sin x)^2)'=2(\sin x) \cos x$ – kingW3 May 27 '17 at 14:57
  • "But why must we change to the second form before we differentiate it": we mustn't. Where did you see that ? –  May 27 '17 at 15:02
  • See What is $\sin^2 x$ equal to? – amWhy May 28 '17 at 17:06
  • "What's the story behind this theory? First of all, there is no theory behind the fact that, as a convention, mathematicians use $\sin^2x$ instead of $(\sin x)^2$ to denote "the square of the (sin x)." Other than that please see E.Josephs answer, and also the answer given by @Mr.Xcoder. Note that the same holds for using $\cos^n x = (\cos x)^n$, and ditto for all the other trig functions. – amWhy May 28 '17 at 17:19

3 Answers3

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$\sin^2(x)$ is just a notation for $(\sin(x))^2$.

So no explanation to give, you usually write $\sin^2 x$ instead of $(\sin (x))^2$ just to avoid the confusion $\sin(x^2)\ne (\sin(x))^2$.

Using the form $(\sin(x))^2$ emphasizes that it is indeed a power, so maybe it is easier to see how to differentiate it.

amWhy
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E. Joseph
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The common notation is $\sin^2(x)$, just to avoid confusion, because $\sin(x)^2$ is ambiguous, and can either be interpreted as: $$\sin (x^2)$$ or $$(\sin x)^2$$

Mr. Xcoder
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I agree with @Mr.Xcoder, but would like to add why you 'have' to

change to the second form before we differentiate it?

It is simply because it is easier to read while doing the calculation: $\frac {d}{dx}[(\sin(x)^2] = 2\sin (x)\frac {d}{dx}[\sin(x)]=2\sin(x)\cos(x)$

Note: you don't 'have' to do this.

Sander B
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