Foundations — Derivatives of eˣ and aˣ — proofs

4.1.19 · D1 · Maths › Calculus I — Limits & Derivatives › Derivatives of eˣ and aˣ — proofs

Isse pehle ki aap Derivatives of eˣ and aˣ — proofs ki ek bhi line follow kar sako, aapko un symbols mein fluent hona hoga jo woh bina ruke istemaal karta hai. Yeh page har ek symbol ko kuch nahi se build karta hai, usi order mein jis order mein woh ek doosre par depend karte hain. Upar se neeche padho; yahan koi bhi symbol neeche define kisi cheez ka istemaal nahi karta.

1. Ek variable $x$ aur ek function $f(x)$

Picture: inputs $x$ ke liye neeche ek number line, aur har input ke upar ek height plot ki jaati hai. Un saari heights ko jodo aur tum ek curve paate ho — function ka graph.

Topic ko yeh kyun chahiye: poora chapter poochta hai "yeh curve kitni steep hai?", toh pehle humein curve khud chahiye. Hamare case mein machine $f(x)=a^x$ ya $f(x)=e^x$ hai.

2. Powers: $a^x$, base, aur exponent

Whole numbers ke liye yeh aasaan hai: $2^3=1\cdot 2\cdot 2\cdot 2=8$.

3. Exponent laws — ek law sab kuch karta hai

Topic ko yeh kyun chahiye: yeh akela splitting trick hi woh reason hai ki ek exponential ka derivative wapas khud ke jaisa aata hai. Parent proof mein, $a^{x+h}=a^x{a}^h$ fixed part $a^x$ ko limit se bahar slide karne deta hai, peeche ek pure number chodta hai.

4. Slope: rise over run

Picture: line par do points chuno. Ek horizontal step (run) khicho aur woh vertical step jo yeh force karta hai (rise). Unka ratio slope hai, aur yeh ek straight line par har jagah same hota hai.

Topic ko yeh kyun chahiye: "exponential kitna steep hai?" yahi question hai "iska slope kya hai?" Lekin ek curve ki steepness point se point par change hoti hai — toh humein ek aur idea chahiye.

5. Limit symbol $\underset{h\to 0}{\lim }$

Topic ko yeh kyun chahiye: parent note mein har derivative ek limit hai, aur §7 ka special constant ek limit se defined hai. Hum ise abhi introduce karte hain, kisi bhi formula ke use se pehle.

6. Tangent line aur derivative $f^{\prime }(x)$

Hum ise kaise compute karte hain (first principles): apna point $x$ chuno, aur ek nearby point $x+h$ jo ek tiny run $h$ door ho. Un do points se guzarne wali line (ek secant) ka slope hai $\frac{f(x+h)-f(x)}{h}=\frac{\text{rise}}{\text{run}}.$ Ab run $h$ ko $0$ ki taraf shrink hone do — §5 mein abhi define kiye limit symbol ka use karke — doosra point pehle mein slide ho jaata hai, aur secant swing karta hai jab tak woh tangent nahi ban jaata. Woh final slope derivative hai, jise $f^{\prime }(x)$ likhte hain.

Topic ko yeh kyun chahiye: yeh formula poore proof ka entry gate hai. Parent note jo bhi derive karta hai woh $a^x$ ko is machine mein feed karke shuru hota hai.

7. Slope-at-zero constant $M(a)$, aur number $e$

Topic ko yeh kyun chahiye: $e$ hi poora point hai — yoh woh base hai jo $e^x$ ko uska apna derivative banata hai.

8. Natural logarithm $\ln a$

Topic ko yeh kyun chahiye: $a=e^{\ln a}$ trick har exponential ko $e$ ke terms mein rewrite karti hai, toh woh ek clean rule ($e^x$ self-derivative hai) unhe sab crack kar sakta hai. Bachne wala factor $\ln a$ nikalta hai — §7 ke mystery constant $M(a)$ ki asli pehchaan.

9. Chain rule (woh ek composition tool)

Topic ko yeh kyun chahiye: $a^x=e^{(\ln a)x}$ likhne ke baad, inner function $u=(\ln a)x$ hai jiska slope $u^{\prime }=\ln a$ hai, aur outer $g(u)=e^u$ hai jiska $g^{\prime }(u)=e^u$ hai. Chain rule us $\ln a$ ko bahar kheenchti hai — $\frac{d}{dx}{a}^x=a^x\ln a$ deliver karti hai.

Pieces topic ko kaise feed karte hain

Map ko ek nadi ki tarah padho jo neeche ki taraf bahti hai: top par sources (ek function, powers, slope, limit) first-principles derivative mein merge hote hain, jo constant $M(a)$ produce karta hai; woh base choose karna jo $M(a)=1$ banata hai define karta hai $e$ ko aur left branch par self-derivative deta hai, jabki $\ln a$ ke through kisi bhi base ko rewrite karna aur chain rule apply karna right branch par general rule deta hai. Dono branches parent note ke do boxed results par land karti hain.

Equipment checklist

Khud ko test karo — tum parent proof ke liye tabhi ready ho jab tum har line ka jawab de sako.

Connections

• Derivatives of eˣ and aˣ — proofs — woh parent topic jiske liye yeh page tumhe prepare karta hai.
• Limits — first principles definition of derivative — derivative formula kahan se aata hai.
• The number e — definitions — $e$ par deeper look.
• Natural Logarithm ln x — $e^x$ ka inverse.
• Chain Rule — parent mein §3 mein istemaal kiya gaya composition tool.
• Power Rule — fixed exponents ke liye contrasting rule.
• Exponential Growth and Decay — jahan self-derivative ek model ban jaata hai, decay $0<a<1$ samait.