Five rules now, and the exam's real question is which one opens a given function. The answer is the outermost operation: the LAST thing done in building $f$. If two pieces were multiplied last, the product rule opens it; divided last, the quotient rule; if one piece was put inside another last, the chain rule; added last, term by term. Whatever sits inside is handled by the rules inside that one. $x^2 \sin x$ is a product: $x^2$ and $\sin x$ side by side. $\sin(x^2)$ is a composition: $x^2$ went INTO the sine. $(\sin x)^2$ is a composition the other way: the sine went into the square. $\dfrac{\sin x}{x}$ is a quotient. Four functions built from the same two pieces, four different first moves, and the trap in the lesson's name is misreading which one you hold.

Brackets and bars are the signposts. A power sits on whatever bracket it touches and nothing else: in $(2x + 1)^3(x - 1)$ the cube belongs to the first bracket, and the two brackets are a product. An exponent of $e$ is everything written upstairs and nothing written beside: $(x + 1)e^{2x}$ is a product whose second factor has $2x$ inside. A log or a root covers everything in its bracket or under its sign: $\ln(x^2 + 1)$ is the log of the whole sum, not $\ln(x^2) + 1$. Read the shape, name the outer rule, then name what sits inside.