You already own every differentiation rule in Unit 3. The last skill is knowing which one to reach for — reading a function’s structure and picking the right procedure before you compute a thing.
Differentiation is parsing. Faced with a messy function, don’t reach for a special case — ask one question: what is the very last operation you’d perform to evaluate it? That operation names your first rule. Apply it, and each piece becomes a smaller sub-problem you attack the same way, until every branch is a basic derivative you already know.
Which rule fires first?
Read the structure, call the outermost rule, then watch the function decompose into a tree and reassemble into \(f'(x)\). The one to study is function 3 — it looks like a product but isn’t.
function
f
What is the outermost operation — the rule that fires first?
The whole decision on one card
Every choice comes down to the top-level structure. Match what you see to the rule it triggers — this single card is the entire unit.
A sum or difference \(f\pm g\)differentiate term by term
A constant multiple \(c\,f\)\(c\,f'\)
A product \(f\cdot g\)\(f'g+fg'\)
A quotient \(f/g\)\(\dfrac{f'g-fg'}{g^2}\)
A composition or power of an inner \(f(g(x)),\ (\ )^{n}\)chain: \(f'(g)\cdot g'\)
A relation you can’t solve for \(y\)implicit differentiation
An inverse value \(\big(f^{-1}\big)'\)\(\dfrac{1}{f'(f^{-1}(x))}\)
Read it, then differentiate
One clean pass through a genuinely mixed function, \(g(x)=e^{x}\ln(x^2+1)\): name the outermost rule, lay down its template, differentiate each piece (one needs the chain rule), then assemble.
The flow on one card
Read the outermost operation — the last thing you’d compute
Lay down that rule’s template, leaving each inner piece unopened
Recurse: ask the same question inside each piece
Stop when every branch is a basic derivative
The core ideas
Read outermost-first
The last operation you’d perform names the first rule you apply.
A power of an inner is a chain
\((\text{stuff})^{n}\) is not the power rule alone — the inner still contributes its derivative.
Recurse
After the outer rule fires, each piece is a fresh sub-problem you solve the same way.
Simplify first when you can
Sometimes the best procedure is algebra: log laws, splitting a fraction, or \(\sqrt{\ }=(\ )^{1/2}\).
Your turn
Quick quiz
Eight fast checks across the whole lesson — identify the rule, spot the trap, pick the efficient path.
Common mistakes & exam tips
Seeing a product inside and stopping there
\((x\sin x)^3\) is chain-first (a power of an inner), product second. The outermost operation is the cube, not the multiplication.
Forcing the quotient rule when splitting is easier
\(\dfrac{x^2+3x}{x}=x+3\) — just differentiate term by term. Reserve the quotient rule for genuine, non-splittable fractions.
Dropping the inner derivative
\(\dfrac{d}{dx}\sin(3x)=3\cos(3x)\), not \(\cos(3x)\). Once the chain fires, the inner’s derivative always tags along.
Treating a constant multiple as a product
\(5x^3\) doesn’t need the product rule; pull the constant out: \(15x^2\). The product rule is for two varying factors.
Exam tips
The “what do you do last?” test picks your first rule every time.