Before the formula, the picture that explains it. Think of \(u\cdot v\) as the area of a rectangle with sides \(u\) and \(v\). When both sides grow a little, how much does the area grow? A strip appears on the right (\(v\cdot du\)) and a strip on top (\(u\cdot dv\)). Add them up — that's the product rule.
What you're looking at: the lime rectangle is \(uv\). Growing both sides adds the gold strip (\(v\,du\)) and the blue strip (\(u\,dv\)). The tiny dashed corner is too small to matter.
Divide by \(dx\) and you have the rule: \(\frac{d}{dx}(uv)=v\frac{du}{dx}+u\frac{dv}{dx}\), usually written \((uv)'=u'v+uv'\). The corner piece \(du\cdot dv\) vanishes in the limit — it's a tiny square next to two long strips.
This is why \((uv)'\ne u'v'\): the product changes when either factor changes, so you need both strips, not their product.
In practice it's a fixed recipe: name the two factors \(u\) and \(v\), differentiate each, then assemble \(u'v+uv'\). Step through \(f(x)=x^2\sin x\) — a case where you can't just expand.
Expand first when both factors are polynomials and multiplying out is clean — \((x+2)(x^2-x)\) is easier expanded. Use the product rule when a factor is trig, exponential, log, or a root, where expanding isn't possible.
\(x^2\sin x\) can't be expanded into a polynomial, so the product rule is the only way in.
Division needs its own rule. For \(\frac{u}{v}\), the derivative is \(\frac{u'v-uv'}{v^2}\). The numerator looks like the product rule but with a minus, and order matters — a popular memory hook is “low d-high minus high d-low, over low squared.”
Unlike the product rule, the quotient rule is not symmetric — swapping the two terms flips the sign and gives the wrong answer. The derivative of the top comes first: \(u'v\), then \(-uv'\).
One restriction: \(v\ne0\). The rule only applies where the denominator isn't zero — which makes sense, since the function itself blows up there.
Last lesson, we told you \(\frac{d}{dx}\tan x=\sec^2x\) but couldn't prove it — we didn't have the quotient rule yet. Now we do. Since \(\tan x=\frac{\sin x}{\cos x}\), the proof falls right out.
The magic step is the Pythagorean identity \(\cos^2x+\sin^2x=1\), which collapses the whole numerator to \(1\). That's how a messy-looking quotient becomes the clean \(\sec^2x\).
The same method derives \(\cot\), \(\sec\), and \(\csc\) — every trig derivative from Lesson 4 traces back to \(\sin\), \(\cos\), and this one rule. Nothing was taken on faith after all.
The quotient rule is powerful but tedious — and often unnecessary. If a fraction simplifies first, do that instead. The smartest students check for cancellation before grinding through the rule. Tap each.
The rule of thumb: look for cancellation first. Only when nothing simplifies — like \(\frac{\sin x}{x^2}\) — do you commit to the full quotient rule.
These four functions each need multiple rules from across the unit — a product or quotient on the outside, power or trig or exponential derivatives on the inside. Pick one and work it through; watch every rule you’ve learned fire in sequence.
Every derivative in calculus — no matter how intimidating — comes apart into pieces like these. The outer structure tells you the rule (product, quotient, and later the chain rule); the inner pieces are the basic derivatives you now know cold.
That’s the whole game. You started the unit computing one derivative with a page of limits. You’re ending it able to differentiate almost anything — in a few clean lines.
A product’s rate of change adds each factor’s contribution: \((uv)′=u′v+uv′\).
The trap: \((uv)′\ne u′v′\). The naive guess ignores that both factors move at once.
The quotient rule, in order: \(\frac{u′v-uv′}{v^2}\). The minus and the order both matter.
Before the quotient rule, check for cancellation. \(\frac{x^2}{x}=x\) needs no rule at all.
Five problems — the product rule on a polynomial-times-exponential and a trig product, the quotient rule, a simplify-first case, and \(\cot x\) the hard way. Try each before revealing.
Eight fast checks across the whole lesson.
The number-one error. The derivative of a product is \(u'v+uv'\), never the product of the derivatives. When in doubt, picture the rectangle strips.
It's \(u'v-uv'\) on top, not \(uv'-u'v\). Swapping them flips every sign. “Low d-high minus high d-low” keeps the order straight.
The quotient rule denominator is \(v^2\), not \(v\). Dropping the square is a silent, costly slip.
If the fraction simplifies — \(\frac{x^2}{x}=x\), \(\frac{x^2+1}{x}=x+\frac1x\) — simplify first. The quotient rule on a simplifiable fraction is wasted effort and extra error-risk.
When \(v=\cos x\), then \(v'=-\sin x\), and that minus interacts with the rule's own minus. Track the signs carefully on quotients involving cosine.
The quotient rule needs \(v\ne0\). Where the denominator is zero, the function and its derivative are both undefined.