Sum, difference, constant multiple rules: Worked examples

4.1.13 · D3 · Maths › Calculus I — Limits & Derivatives › Sum, difference, constant multiple rules

Yahan sab kuch parent note ke teen facts par tikaa hai. Ek bhi symbol use karne se pehle unhe seedhe shabdon mein restate karte hain:

Hum individual pieces ke liye Power rule par bhi rely karte hain: $\frac{d}{d x} [x^{n}] = n x^{n - 1}$ — padho as "exponent ko neeche front mein lao, phir exponent ko ek se kam karo". Baaki sab inhe combine karna hai.

The scenario matrix

Neeche har class ka problem hai jo linearity present kar sakti hai. Har row ek "cell" hai; aage jo examples hain unpar cell tag laga hai.

#	Cell (case class)	Isme tricky kya hai	Example
A	Plain polynomial, saare positive terms	warm-up, power rule + sum	Ex 1
B	Mixed signs / multi-term group ka subtraction	minus sign distribute karna zaroori hai	Ex 2
C	Constant term + constant multiple	constant ki derivative $0$ hoti hai	Ex 1, Ex 2
D	Negative & fractional exponents (pehle rewrite karo)	roots aur $1/ x$ powers ki tarah chhupi hain	Ex 3
E	Alag function types milaaye hue (sin, $e^{x}$ , ln)	linearity inhe saath rehne deti hai	Ex 4
F	Degenerate: constant multiple $c = 0$ ya $c = 1$	$c$ ki limiting/edge values	Ex 5
G	Ek specific point par slope evaluate karo (slope ka sign)	number daalo, geometry padho	Ex 6
H	Real-world word problem (rates add hoti hain)	words → linearity mein translate karo	Ex 7
I	Exam twist: product jaisa lagta hai, hai nahi	differentiate karne se pehle expand karna zaroori	Ex 8

The worked examples

Worked example Example 1 — Cell A + C: plain polynomial ek constant ke saath

$y = 2 x^{3} + 5 x^{2} + 4 x + 9$ differentiate karo.

Forecast: aage padhne se pehle $y^{'}$ guess karo. Kitne terms bachenge? $9$ ka kya hoga?

Sum split karo: $y^{'} = \frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [5 x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [9]$ . Yeh step kyun? Sum rule hume allow karta hai har term ko apne aap differentiate karne aur add karne ka — koi term doosre se interact nahi karta.
Constants bahar nikalo aur power rule term by term apply karo: $y^{'} = 2 (3 x^{2}) + 5 (2 x) + 4 (1) + 0 = 6 x^{2} + 10 x + 4.$ Yeh step kyun? Constant multiple rule $2$ aur $5$ ko bahar rehne deta hai; power rule har $x^{n}$ ko handle karta hai; last term $9$ ek flat line hai isliye uski slope $0$ hai.

Verify: constant $9$ gayab ho gaya (ek horizontal line ki steepness zero hoti hai ✅), aur har exponent ek se kam hua, power rule se match karta hai.

Worked example Example 3 — Cell D: negative aur fractional exponents (pehle rewrite karo)

$y = \frac{4}{x ^{2}} + 6 x - \frac{x}{3}$ differentiate karo.

Forecast: power rule tab tak use nahi kar sakte jab tak har piece $c x^{power}$ ki tarah likhi na ho. Teen powers kya hain?

Har piece ko ek scaled power ki tarah rewrite karo: $y = 4 x^{- 2} + 6 x^{1/2} - \frac{1}{3} x .$ Yeh step kyun? $\frac{4}{x ^{2}} = 4 x^{- 2}$ ( $x^{2}$ se divide karna $x^{- 2}$ se multiply karna hai); $x = x^{1/2}$ ; aur $\frac{x}{3} = \frac{1}{3} x$ . Power rule ko explicit exponent chahiye, isliye hum ek banate hain.
Power rule se har ek differentiate karo, constants bahar ride karte hue: $y^{'} = 4 (- 2) x^{- 3} + 6 (\frac{1}{2}) x^{- 1/2} - \frac{1}{3} (1) = - 8 x^{- 3} + 3 x^{- 1/2} - \frac{1}{3} .$
Readability ke liye root/fraction form mein wapas rewrite karo: $y^{'} = - \frac{8}{x ^{3}} + \frac{3}{x} - \frac{1}{3} .$ Yeh step kyun? Negative exponents matlab "denominator mein"; $- 1/2$ power matlab "square root ka ulta".

Verify: exponent $- 2$ gira $- 3$ par aur $1/2$ gira $- 1/2$ par — har ek exactly ek se kam hua, jaisa power rule demand karta hai.

Worked example Example 5 — Cell F: degenerate constants

$c = 0$ aur $c = 1$ Differentiate karo (a) $y = 0 \cdot x^{5} + 3 x$ aur (b) $y = 1 \cdot cos x - x^{2}$ .

Forecast: (a) mein, kya $x^{5}$ term ka koi matlab hai?

Part (a): constant multiple rule with $c = 0$ deta hai $\frac{d}{d x} [0 \cdot x^{5}] = 0 \cdot (5 x^{4}) = 0$ . Yeh step kyun? Ek function ko $0$ se scale karna zero function produce karta hai, jiski slope har jagah $0$ hoti hai. Toh $x^{5}$ term derivative ke liye invisible hai. $y^{'} = 0 + 3 = 3.$
Part (b): constant multiple rule with $c = 1$ derivative ko unchanged chhodata hai: $\frac{d}{d x} [1 \cdot cos x] = 1 \cdot (- sin x) = - sin x$ . $y^{'} = - sin x - 2 x .$ Yeh step kyun? $1$ se multiply karna kuch nahi badalta — edge case $c = 1$ identity hai.

Verify: $c = 0$ ek term delete karta hai, $c = 1$ use preserve karta hai — constant multiple rule ki do extreme values exactly waise behave karti hain jaisa ek scaling factor karta hai.

Worked example Example 6 — Cell G: ek point par slope ka sign padhna (geometric)

$y = x^{3} - 3 x$ ke liye $y^{'}$ nikalo, phir $x = - 2$ , $x = 0$ , aur $x = 1$ par slope evaluate karo. Har jagah curve ki steepness ka sign kya hai?

Forecast: $x = 0$ par curve rising hai, falling hai, ya flat? Graph imagine karke guess karo.

Linearity + power rule se differentiate karo: $y^{'} = 3 x^{2} - 3.$ Yeh step kyun? Sum/difference do terms split karta hai; power rule + constant multiple har ek handle karta hai.
Har point mein plug karo:
- $x = - 2 : y^{'} = 3 (4) - 3 = 9$ → positive slope (rising).
- $x = 0 : y^{'} = 3 (0) - 3 = - 3$ → negative slope (falling).
- $x = 1 : y^{'} = 3 (1) - 3 = 0$ → zero slope (flat spot / turning point). Yeh step kyun? Derivative khud ek function hai; usmein $x$ daalo toh wahan ki steepness milti hai.

Verify: figure dikhata hai curve left mein rising hai (slope $+ 9$ ), $x = 1$ ke paas ek flat point se guzarti hai (slope $0$ ), aur beech mein falling hai (slope $- 3$ ) — slope ke teeno signs appear karte hain, hamare numbers se match karte hue.

Worked example Example 7 — Cell H: ek real-world word problem (rates add hoti hain)

Ek tank do pipes se bhara ja raha hai. Pipe A paani add karta hai $A (t) = 4 t^{2}$ litres ki position par aur pipe B $B (t) = 3 t$ litres par, jahan $t$ minutes mein hai. Total paani hai $W (t) = A (t) + B (t)$ . $t = 2$ minutes par tank kitni tezi se bhar raha hai?

Forecast: guess karo — kya total fill rate sirf A ki rate plus B ki rate hai?

"Kitni tezi se" matlab derivative $W^{'} (t)$ . Sum rule se, total ki rate individual rates ka sum hai: $W^{'} (t) = A^{'} (t) + B^{'} (t) .$ Yeh step kyun? Yeh linearity ka physical meaning hai: combined rates add hoti hain — exactly woh do-runners wali picture parent note se.
Har ek differentiate karo: $A^{'} (t) = 8 t$ aur $B^{'} (t) = 3$ , toh $W^{'} (t) = 8 t + 3$ .
$t = 2$ par evaluate karo: $W^{'} (2) = 8 (2) + 3 = 19$ litres per minute.

Verify (units): $8 t$ ke units hain litres/min (litres $4 t^{2}$ se, min $d / d t$ se) aur $3$ bhi likewise — litres/min ko litres/min mein add karna deta hai litres/min, ek valid rate. Numeric: $19$ L/min. ✅

Worked example Example 8 — Cell I: exam twist jo product

jaisa lagta hai $y = (x + 2) (x - 5)$ differentiate karo.

Forecast: yeh ek product hai — kya tum product rule pakdoge? Kya pakdna chahiye?

Pehle expand karo (linearity sirf sums handle karta hai, products nahi): $y = x^{2} - 5 x + 2 x - 10 = x^{2} - 3 x - 10.$ Yeh step kyun? Parent note warn karta hai ki product ki derivative products of derivatives nahi hoti. Lekin expand karne ke baad, $y$ ek plain sum hai, aur linearity apply hoti hai. (Alternatively Product rule use karo — same answer milega.)
Sum differentiate karo: $y^{'} = 2 x - 3.$

Verify: Product rule $(f g)^{'} = f^{'} g + f g^{'}$ se cross-check karo $f = x + 2, g = x - 5$ ke saath: $(1) (x - 5) + (x + 2) (1) = x - 5 + x + 2 = 2 x - 3$ . ✅ Same answer — trap yeh sochna tha ki tum linearity use nahi kar sakte; expand karne ne ise legal bana diya.

Recall Self-check: kaun sa cell sabse mushkil tha?

Sign trap (Cell B) ::: leading minus har term par bracket ke andar distribute hona chahiye. Rewrite trap (Cell D) ::: roots aur fractions ko power rule se pehle explicit powers mein convert karo. Product trap (Cell I) ::: pehle expand karo — linearity kabhi product split nahi karta.

Connections

Sum, difference, constant multiple rules — parent rules jo yahan drill kiye gaye.
Power rule — per-term engine jo almost har example mein use hua.
Product rule — sibling rule jo Cell I mein chahiye.
Definition of the derivative (limit of difference quotient) — slopes kahan se aati hain.
Limit laws (sum, scalar, product of limits) — sum rule kyun legal hai.
Linearity of integration — same trick, ulta chalaya.
Linear operators — abstract reason ki alag function types kabhi interfere kyun nahi karte.