Choosing the procedure AB & BC

Every expression has one outermost operation, and that operation picks the rule. Find it by asking what you would do last if you were evaluating at a number:

1. Last step is an addition or subtraction, so differentiate term by term.
2. Last step is a multiplication of two non-constant factors, so use the product rule.
3. Last step is a division by a non-constant, so use the quotient rule.
4. Last step is applying a function to an inside expression, so use the chain rule.
5. Last step is multiplying by a constant, so pull the constant out and differentiate what remains.

Only after the top level is named does an inner piece get its own diagnosis. A single expression may need three rules, but they enter in a definite order, outermost first.

The most frequent misdiagnosis is treating a sum as though a rule applied to it as a whole. There is no product rule for $x^2 + \sin x$; it is simply $2x + \cos x$. Rules attach to operations, and addition already distributes over differentiation.

A rule that can be applied is not always the rule that should be. Three rewrites pay for themselves constantly:

1. Constant denominators. $\frac{x^3}{5}$ is $\frac{1}{5}x^3$, so the derivative is $\frac{3x^2}{5}$. The quotient rule works but wastes a step and invites sign errors, because 5 is a constant and has derivative zero.
2. Radicals to exponents. $\sqrt[3]{x}$ is $x^{1/3}$, so the power rule applies directly. Leaving it as a radical leaves no rule that fits.
3. Split numerators. $\frac{x^2 + 3x}{x}$ simplifies to $x + 3$ for $x \ne 0$, with derivative 1. Reaching for the quotient rule here is several lines of work to reach the same 1.

The opposite error is also real: expanding $(x^2+1)^5$ by hand to avoid the chain rule turns a two-factor answer into a sixth-degree polynomial. Simplify when it removes work, not as a reflex.

When a factor of a product is itself a composition, both rules run, with the product rule at the top:

$$\frac{d}{dx}\left[x^2\sin(3x)\right] = 2x\sin(3x) + x^2\cdot 3\cos(3x).$$

The product rule chose the two-term shape; the chain rule supplied the 3 inside the second term. Neither substitutes for the other.

Quotients of composites work the same way. For $\frac{\sin(2x)}{x^2+1}$ the quotient rule governs, and the chain rule handles the numerator's derivative $2\cos(2x)$. A useful check on a finished answer: count the rules the structure demanded and confirm each one left a visible trace. A missing inner factor is the trace that goes missing most often.

Several routes are often valid, and on a timed exam the cheapest one matters. For $y = \frac{1}{x^2+1}$ you may use the quotient rule, or rewrite as $(x^2+1)^{-1}$ and use the chain rule. Both give $\frac{-2x}{(x^2+1)^2}$; the second is shorter and has fewer places to drop a sign.

For $y = \ln\left(x^2\sqrt{x+1}\right)$, differentiating directly means a chain rule wrapped around a product wrapped around a radical. Expanding the logarithm first gives $2\ln x + \frac{1}{2}\ln(x+1)$, and the derivative is $\frac{2}{x} + \frac{1}{2(x+1)}$ in one line. Recognizing that a logarithm turns products into sums is procedure selection, not algebra trivia.