I'm new to calculus and self-learning it, I am having trouble grasping why $\sin^2 5x$ can be rewritten as $(\sin 5x)^2$
I fumble when I try to understand the logic behind it. For example, if I plug in a number for 'sin' I would get a different answer when I plugged it in the original equation and when I plugged it into the rewritten equation. Just trying to understand the logic and the 'why', so I don't go through calculus mindlessly solving equations. Thanks.
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$\begingroup$Welcome to Math SE!
The question you have is just one of notation. It's pretty standard to write, generally, $[\sin(x)]^n$ as $\sin^n(x)$. The latter notation is probably more common than the former, in fact.
This notation holds broadly for integer values of $n$ except in the case of $n=-1$, when $\sin^{-1}(x)$ denotes $\arcsin(x)$. If you want to write $[\sin(x)]^{-1}$, you could write it as $[\sin(x)]^{-1}$, $\frac{1}{\sin(x)}$, or $\csc(x)$.
So, when you need to take the derivative of $\sin^2(5x)$, it may be a bit easier to think of it as $[\sin(5x)]^2$, then apply the chain rule. They both mean the same thing, it's just easier to recognize the ordering of the function composition: $5x$ is the input to $\sin(\cdot)$, and $\sin(\cdot)$ is the input to $(\cdot)^2$.
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