The rules for sums, products and quotients are not conventions to memorise; each is a short argument with the difference quotient, and the argument is worth seeing once because it explains the shape of the answer. The product rule looks strange until the trick is seen: add and subtract a middle term, so that one factor is held still while the other moves.

The area of a rectangle whose two sides both grow changes by the first side times the growth of the second, plus the second side times the growth of the first, plus a corner that is negligible. The product rule is that sentence, and the corner is what vanishes in the limit.

If $f$ and $g$ are differentiable at $c$ then so are $f\pm g$, $kf$ and $fg$, and so is $\tfrac fg$ provided $g(c)\ne0$, with $(f\pm g)'=f'\pm g'$, $(kf)'=kf'$, $(fg)'=f'g+fg'$ and $\big(\tfrac fg\big)'=\tfrac{f'g-fg'}{g^2}$. Every one is proved from the difference quotient and the algebra of limits, and the product rule needs one extra move.

• $(x^n)'=nx^{n-1}$ for every natural $n$, by induction from the product rule.

Write the difference quotient of the product and insert a middle term that is subtracted and added again — the one obtained by moving only one of the two factors. The quotient then splits into two pieces, each of which is a difference quotient of one function multiplied by a value of the other. Taking limits, the two difference quotients become derivatives, one factor is evaluated at the point, and the other is evaluated at a moving point, which converges because differentiability brings continuity with it.

Write the difference quotient of the product.

Subtract and add a middle term in which only one factor has moved.

Group and factor: each piece is one difference quotient times a value of the other function.

The second function is continuous at the point, because it is differentiable there.

The algebra of limits applied to the two pieces.

The middle term moves one factor and holds the other, which is why the answer has two terms.

• Sums, constant multiples, products and quotients all have rules, each proved from the difference quotient.
• The product rule has two terms because the proof moves one factor at a time.
• The quotient rule follows from the product rule and needs a non-zero denominator.
• $(x^n)'=nx^{n-1}$ follows by induction.

Great work! You now know how to:

• State and use the sum, constant, product and quotient rules
• Prove the product rule by inserting a middle term
• Derive the quotient rule from the product rule
• Establish the power rule by induction