Derivative from first principles — difference quotient definition

4.1.10 · Maths › Calculus I — Limits & Derivatives

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HUM yeh kyun seekhte hain?

Tum already jaante ho seedhi line ki slope kaise nikaalte hain: do points lo, "rise over run" karo. Simple.

Lekin $y=x^2$ jaisi curve ki har point pe alag steepness hoti hai. $x=1$ pe slope gentle hai; $x=5$ pe steep hai. "Rise over run" ke liye do points chahiye — toh hum ek point pe slope kaise measure karein?

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Difference Quotient KYA hota hai?

• Numerator $f(x+h)-f(x)$ = rise (output mein change).
• Denominator $h$ = run (input mein change).

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Isko compute kaise karein — algorithm

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Worked Example 1 — $f(x)=x^2$

Goal: $f^{\prime }(x)$ first principles se nikaalein.

Step 1 — $f(x+h)$ likho. $f(x+h)=(x+h{)}^2=x^2+2xh+h^2$ Yeh step kyun? Paas wale point pe output jaanna zaroori hai taaki hum compare kar sakein.

Step 2 — numerator. $f(x+h)-f(x)=(x^2+2xh+h^2)-x^2=2xh+h^2$ Yeh step kyun? $x^2$ terms cancel ho jaate hain — yahi cancellation hai jis wajah se ek finite slope exist karta hai.

Step 3 — $h$ se divide karo, factor karo aur cancel karo. $\frac{2xh+h^2}{h}=\frac{h(2x+h)}{h}=2x+h$ Yeh step kyun? $h$ factor out karna $\frac{0}{0}$ ki problem hatata hai. Ab expression $h=0$ pe safe hai.

Step 4 — limit lo. $f^{\prime }(x)={\lim }_{h\to 0}(2x+h)=2x$ Yeh step kyun? Jaise $h\to 0$, bacha hua $h$ gayab ho jaata hai. Answer: $f^{\prime }(x)=2x$.

Sanity check: $x=3$ pe, slope $=6$ (steep); $x=0$ pe, slope $=0$ (parabola ka bottom flat hai). ✓

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Worked Example 2 — $f(x)=\frac{1}{x}$

Step 1. $f(x+h)=\frac{1}{x+h}$.

Step 2 — numerator (common denominator). $\frac{1}{x+h}-\frac{1}{x}=\frac{x-(x+h)}{x(x+h)}=\frac{-h}{x(x+h)}$ Yeh step kyun? Fractions subtract karne ke liye common denominator chahiye — isse hidden $h$ saamne aata hai.

Step 3 — $h$ se divide karo. $\frac{1}{h}\cdot \frac{-h}{x(x+h)}=\frac{-1}{x(x+h)}$ Yeh step kyun? $h$ cancel ho jaata hai, $\frac{0}{0}$ khatam ho jaata hai.

Step 4 — limit. $f^{\prime }(x)={\lim }_{h\to 0}\frac{-1}{x(x+h)}=\frac{-1}{x^2}$ Answer: $f^{\prime }(x)=-\frac{1}{x^2}$. (Hamesha negative — $1/x$ jahan defined hai wahan hamesha neeche slope karta hai.) ✓

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Worked Example 3 — $f(x)=\sqrt{x}$ ("conjugate" trick)

Step 1–2. $\sqrt{x+h}-\sqrt{x}$ — stuck lag raha hai, factor karne ke liye koi $h$ nahi.

Move yeh hai: conjugate $\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}}$ se multiply karo.

=\frac{(x+h)-x}{h\big(\sqrt{x+h}+\sqrt{x}\big)}=\frac{h}{h(\cdots)}$$ *Yeh step kyun?* $(\sqrt a-\sqrt b)(\sqrt a+\sqrt b)=a-b$ — yeh numerator mein $h$ *banata* hai taaki woh cancel ho sake. **Step 3–4.** $$=\frac{1}{\sqrt{x+h}+\sqrt{x}}\ \xrightarrow{h\to 0}\ \frac{1}{2\sqrt{x}}$$ **Answer: $f'(x)=\dfrac{1}{2\sqrt{x}}$.** ✓ --- ## Pehle Forecast Karo, Phir Verify Karo > [!example] Compute karne se pehle predict karo > Answer padhne se pehle: $f(x)=3x+5$ (ek seedhi line) ke liye, **forecast** karo $f'(x)$ kya hoga. > **Forecast:** slope constant hai $=3$. > **Verify:** $\dfrac{[3(x+h)+5]-[3x+5]}{h}=\dfrac{3h}{h}=3 \to 3$. ✓ Ek line ka "first-principles" slope bas uska slope hota hai. > [!mistake] Steel-man: "$f'$ aslmein $f$ ki value hai, bas chhoti." > Lagta hai sahi hai kyunki $x^2$ differentiate karne pe $2x$ aaya jo "related" lagta hai. **Lekin** derivative **rate of change** measure karta hai, size nahi. $f(10)=100$ jabki $f'(10)=20$ — bilkul alag cheezein hain. **Fix:** $f'$ ko hamesha "slope / kitni tezi se" padhein, kabhi "ek chhota copy of $f$" mat samjhein. > [!mistake] $x+h$ *har jagah* substitute karna bhool jaana. > Students $f(x)=x^2$ ke liye $f(x+h)=x^2+h$ likh dete hain. Galat — har $x$ $(x+h)$ ban jaata hai: $(x+h)^2$. **Fix:** $x+h$ ko ek block ki tarah treat karo aur carefully expand karo. --- ## #flashcards/maths $f$ ka difference quotient kya hai? ::: $\dfrac{f(x+h)-f(x)}{h}$, secant line ki slope. Derivative ko first principles se define karo. ::: $f'(x)=\lim_{h\to0}\dfrac{f(x+h)-f(x)}{h}$ Difference quotient mein $h=0$ turant kyun nahi set kar sakte? ::: Isse $\tfrac{0}{0}$ (undefined) milta hai; pehle $h$ algebraically cancel karna padta hai, phir limit lena padta hai. Geometrically, derivative kiske barabar hota hai? ::: Us point pe curve ki tangent line ki slope ke barabar. Jaise $h\to 0$, secant line ___ line ban jaati hai. ::: tangent $f(x)=x^2$ ke liye $f'(x)$ derive karo. ::: $\lim_{h\to0}\frac{2xh+h^2}{h}=\lim_{h\to0}(2x+h)=2x$ $f(x)=\sqrt{x}$ ko first principles se handle karne ki kya trick hai? ::: Conjugate $\sqrt{x+h}+\sqrt{x}$ se multiply karo taaki cancel karne ke liye $h$ create ho sake; answer $\frac{1}{2\sqrt x}$. $1/x$ ka derivative first principles se? ::: $-1/x^2$ Numerator $f(x+h)-f(x)$ kya represent karta hai? ::: Rise (output mein change). --- > [!recall]- Feynman: 12-saal ke bachche ko samjhao > Socho tum ek car chalaa rahe ho. Tumhari **position** time ke saath badlti hai. Ek exact instant pe tumhari **speed** jaanne ke liye, tum sirf ek clock-tick nahi dekh sakte — speed ke liye ek *change* chahiye. Toh tum check karte ho ki thodi si der pehle tum kahan the aur ab kahan ho, aur distance ko time se divide karte ho. Us "thodi si der" ko chhota se chhota karte jao, aur answer tumhari **exact speed right now** par settle ho jaata hai. Derivative wahi settling-down number hai. Curve ke liye, "speed" ka matlab hai "steepness" — aur hum isse measure karte hain ek bahut paas wala doosra point chun ke, unke beech ki line ki slope nikaal ke, phir us doosre point ko tab tak slide karte hain jab tak woh touch na karein. > [!mnemonic] Recipe yaad karo > **"Plug, Subtract, Cancel, Shrink"** — **P**lug in $x+h$, **S**ubtract $f(x)$, **C**ancel the $h$, **S**hrink $h\to 0$. --- ## Connections - [[Limits — formal definition and one-sided limits]] — derivative aslmein ek limit hi hai; yeh uska parent idea hai. - [[Indeterminate forms 0 over 0]] — explain karta hai ki substitute karne se pehle cancel kyun karna zaroori hai. - [[Secant and Tangent lines]] — algebra ke peeche geometric picture. - [[Power Rule]] — ek shortcut jo innati first principles se *prove* hota hai ($\frac{d}{dx}x^n=nx^{n-1}$). - [[Continuity and Differentiability]] — differentiable $\Rightarrow$ continuous, lekin ulta nahi. - [[Average vs Instantaneous Rate of Change]] — secant = average, tangent = instantaneous. ## 🖼️ Concept Map ```mermaid flowchart TD SLOPE[Slope needs two points] CURVE[Curve has different slope everywhere] TRICK[Second point x plus h] DQ[Difference quotient] SECANT[Secant line slope] LIMIT[Take limit as h approaches 0] DERIV[Derivative f prime x] TANGENT[Tangent line slope] ZERO[Cannot set h=0 first gives 0/0] RECIPE[4-step recipe factor out h] SLOPE -->|fails on| CURVE CURVE -->|solved by| TRICK TRICK -->|forms| DQ DQ -->|measures| SECANT DQ -->|apply| LIMIT ZERO -->|forces| RECIPE RECIPE -->|cancels h then| LIMIT LIMIT -->|yields| DERIV DERIV -->|equals slope of| TANGENT SECANT -->|rotates into| TANGENT ``` ## 🔬 Deep Dive > [!intuition] Aur gehrai mein jao — visual, zero se > Is topic ke step-by-step 3Blue1Brown-style breakdowns. - [[4.1.10 D1 Foundations|D1 · Foundations — har symbol zero se]] - [[4.1.10 D2 Visual Walkthrough|D2 · Visual walkthrough — derivation pictures mein]] - [[4.1.10 D3 Worked Examples|D3 · Worked examples — har scenario]] - [[4.1.10 D4 Exercises|D4 · Exercises — graded, full solutions ke saath]] - [[4.1.10 D5 Question Bank|D5 · Question bank — concept traps]]