First PrinciplesStart anywhere. Prove it, then move on.

Rung 21

Rules of differentiation

The handful of patterns that cover almost everything.

Best attempted after 20. The derivative. Nothing stops you trying this now — the gate will tell you if you were right.

The gate

Differentiate a mixed set of functions using the power, product, and quotient rules, then verify every one numerically against the limit definition. Then derive the product rule itself from the definition — a rule you have proved is a rule you can rebuild when you misremember it.

Nobody checks this but you. Do it honestly and the rungs above hold; do it loosely and they will not, somewhere further up where the cause is much harder to find.

Deriving every derivative from the limit definition is correct and unbearable. The rules exist so you do not have to — each one a limit computation somebody did once, in general, so nobody need do it again.

They are labour-saving devices, not axioms. That distinction matters when one of them slips out of memory.

Why this is on the ladder

Because rung 44 will have you writing derivatives of activation functions by hand, and you need them to be quick and automatic. Fluency here is what makes the chain rule at rung 22 feel like an application rather than a new mountain.

Do this

Learn the power rule, the product rule, and the quotient rule. Then, for each function you differentiate, verify numerically using rung 20's method — the symbolic answer and the numerical slope should agree to several decimals.

That verification habit is not busywork. It is the same move as rung 5's substituting back and rung 16's double count: an independent check that costs a minute and catches the errors staring cannot.

Then derive the product rule from the limit definition. The trick is adding and subtracting the same term in the numerator so it splits into two recognisable pieces. Work at it before looking it up; the struggle is what makes it stick.

Where people get stuck

Assuming the derivative of a product is the product of the derivatives. It is not, and a single numerical check kills the idea immediately — which is precisely why the check is in the gate.

The quotient rule's sign and order get scrambled constantly. Rather than memorising harder, rewrite the quotient as a product with a negative power and use the product rule. Fewer things to remember is usually the better trade.

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