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.