What’s the difference between the notation $(sin(x))^\prime$ and $sin^\prime (x)$? What’s the difference between the notation $(\sin(x))^\prime$ and $\sin^\prime (x)$? Where prime denote the derivative of some function. This question seems trivial at first. If we change the argument of function to something different expression of x. The outcome becomes completely different. To give an example, consider the function $\sin(\frac{1}{x})$. We have $(\sin(\frac{1}{x}))^\prime=\sin^\prime(\frac{1}{x})  (\frac{1}{x})^\prime = - \frac{1}{x^2} \cos(\frac{1}{x})$. In this example, we treat $\sin$ function as composite function. We are usually custom to write $(f(x))^\prime$ as $f^\prime (x)$. Perhaps we can not apply this “notation” to this given function. Although, I don’t known the precise  definition of $\sin$ function, I am assuming (due to lack of my knowledge) that, it is defined on $\mathbb{R}$, and it’s range is $[-1,1]$. Since, $(\sin(x))^\prime$ and $\sin^\prime (x)$ are equal for $x \in \mathbb{R}$. Isn’t $\frac{1}{x} \in \mathbb{R}$ is also a real number? So, why do we have different outcome for derivative operation? Is $\sin$ function is really a composite function?
Warning: My question generally contains error and/or misunderstanding of concept. If you find something wrong then mention it on comment.
 A: Technically, $[\sin(x)]'$ is an abuse of notation, since $\sin(x)$ is not itself the function being differentiated, but the output of the function being differentiated: it is an element in the range of the function. So, in any case, if you want to embrace the pedantry, you should use the notation $\sin'(x)$ instead, because now it is clear that you are evaluating the function $\sin'$ at $x$ to obtain the output $\sin'(x)$.
However, there is another reason why you should keep in mind a distinction between $[\sin(x)]'$ and $\sin'(x)$. Suppose we naively replace $x$ with $f(x)$ instead. Now, one would typically interpret $[\sin(f(x))]'$ as an expression where the chain rule is applied: hence you would simplify this as $\cos[f(x)]f'(x)$. But you would never interpret $\sin'(f(x))$ as $\cos[f(x)]f'(x)$. Instead, you would interpret it as $\cos[f(x)]$, since you are taking $\sin'=\cos$, and you are taking $f(x)$ as the input. Because of these shenanigans, you can actually write the equation $$(\sin[f(x)])'=\sin'[f(x)]f'(x),$$ and this would be technically correct, even though it definitely does not look correct. So the fact is that a distinction is intended to exist, but it works in a way that is extremely misleading. This is the other reason why you should avoid using the notation $[f(x)]'$ and stick to $f'(x)$ instead. And if you really need a notation that allows you to use the chain rule, then instead of writing $(\sin[f(x)])',$ you should write $(\sin\circ{f})'(x)$ to avoid confusion. This makes the distinction between $\sin'[f(x)]$ and $(\sin\circ{f})'(x)$ clear and without any abuse of notation. Hence you can write $$(\sin\circ{f})'(x)=\sin'[f(x)]f'(x),$$ and now the chain rule as applied makes sense visually.
A: You correctly note that the use of the prime to indicate differentiation is ambiguous in this context. I've never seen it and you should not use it. If you actually encounter it somewhere, hope the context makes the author's meaning clear.
