Quotient rule — proof — Maths notes

The statement

Derivation from first principles

We use the limit definition: $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.$

Step 1 — write the difference quotient. $f(x+h)-f(x)=\frac{u(x+h)}{v(x+h)}-\frac{u(x)}{v(x)}.$ Why this step? This is literally the definition; everything else is algebra to make the limit computable.

Step 2 — combine over a common denominator. $=\frac{u(x+h)\,v(x)-u(x)\,v(x+h)}{v(x+h)\,v(x)}.$ Why this step? A single fraction is easier to manipulate, and it isolates the denominator $v(x+h)v(x)$ which will become $v^2$ in the limit.

Step 3 — the "add and subtract zero" trick. Insert $-u(x)v(x)+u(x)v(x)=0$ into the numerator: $u(x+h)v(x)-u(x)v(x+h) = \underbrace{\big[u(x+h)v(x)-u(x)v(x)\big]}_{\text{change in }u} - \underbrace{\big[u(x)v(x+h)-u(x)v(x)\big]}_{\text{change in }v}.$ Why this step? We engineer difference quotients for $u$ and $v$ separately. Factor: $= v(x)\big[u(x+h)-u(x)\big] - u(x)\big[v(x+h)-v(x)\big].$

Step 4 — divide by $h$ . Recall $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ , so divide the whole thing by $h$ : $\frac{f(x+h)-f(x)}{h}=\frac{1}{v(x+h)v(x)}\left[v(x)\,\frac{u(x+h)-u(x)}{h} - u(x)\,\frac{v(x+h)-v(x)}{h}\right].$ Why this step? Now two recognizable difference quotients appear: they are the definitions of $u'(x)$ and $v'(x)$ .

Step 5 — take the limit $h\to 0$ . Use that $v$ is differentiable (hence continuous), so $v(x+h)\to v(x)$ : $f'(x)=\frac{1}{v(x)\,v(x)}\Big[v(x)\,u'(x)-u(x)\,v'(x)\Big]=\frac{u'(x)v(x)-u(x)v'(x)}{\big(v(x)\big)^2}.\qquad\blacksquare$ Why continuity matters: without $v(x+h)\to v(x)$ , the denominator $v(x+h)v(x)$ wouldn't tend to $v(x)^2$ and the whole argument collapses.

Worked examples

Common mistakes (steel-manned)

Flashcards

State the quotient rule for

(u/v)'

\dfrac{u'v - u v'}{v^2}

In the proof, what algebraic trick creates the two difference quotients?

Add and subtract

u(x)v(x)

in the numerator ("add zero").

Why must

v

be continuous for the proof's last step?

v(x+h)\to v(x)

, making the denominator tend to

v(x)^2

Why is there a minus sign in the quotient rule?

Because growing the denominator shrinks the fraction; differentiating

1/v

gives

-v'/v^2

Derive

\frac{d}{dx}\tan x

via the quotient rule.

\frac{\cos^2x+\sin^2x}{\cos^2x}=\sec^2x

What special case do you get from

u=1

Reciprocal rule

\frac{d}{dx}(1/v)=-v'/v^2

Quick test that kills "

u'/v'

": which function?

f=x^2/x=x

has

f'=1

, but

u'/v'=2x

— contradiction.

Recall Feynman: explain to a 12-year-old

Imagine sharing pizza: top = number of slices you have, bottom = number of friends sharing. If you get more slices (top grows), everyone's share grows — that's the $+u'v$ part. If more friends show up (bottom grows), everyone's share shrinks — that's the $-uv'$ part (minus!). And the bigger the crowd already is, the less each extra friend changes things — that's why we divide by the crowd size squared, $v^2$ .

Connections

Product rule — proof — same "add-zero" trick; quotient rule = product rule of $u$ and $1/v$ .
Chain rule — needed to see $1/v$ as $v^{-1}$ giving $-v'/v^2$ .
Limit definition of derivative — the engine of this whole proof.
Continuity — justifies $v(x+h)\to v(x)$ .
Derivatives of trig functions — $\tan,\cot,\sec,\csc$ all come from this rule.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, quotient rule tab use hota hai jab ek function doosre se divide ho raha ho, jaise $\frac{u}{v}$ . Bahut log galti karte hain ki top aur bottom ko alag-alag differentiate kar dete hain — ye galat hai. Sahi formula hai: "Lo dee-Hi minus Hi dee-Lo, over Lo-Lo", yaani $\frac{u'v - uv'}{v^2}$ .

Proof ka asli jaadu ek trick hai: limit definition se shuru karo, common denominator banao, aur phir numerator mein $u(x)v(x)$ ko add-subtract kar do (zero add karna). Isse magically do alag difference quotients ban jaate hain — ek $u'$ ke liye, ek $v'$ ke liye. Phir $h\to 0$ limit lete hain, aur kyunki $v$ continuous hai, $v(x+h)\to v(x)$ ho jaata hai, jisse denominator $v^2$ ban jaata hai.

Minus sign kyun? Socho pizza ko friends mein baant rahe ho. Zyada slices (top badha) toh sabka share badha — plus. Lekin zyada friends aa gaye (bottom badha) toh sabka share kam ho gaya — isliye minus. Aur jitni badi crowd, utna kam farak — isliye $v^2$ se divide.

Ye rule yaad rakhna important hai kyunki $\tan x, \sec x$ jaise saare trig derivatives isi se nikalte hain. Mnemonic ratlo aur ek baar khud derive karke dekho — exam mein kabhi nahi bhoologe.

Quotient rule — proof

The statement

Derivation from first principles

Worked examples

Common mistakes (steel-manned)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Calculus I — Limits & Derivatives

Connections