Continuity & Differentiability 📈

Kab ek function "smooth" hai? Kab differentiate kar sakte hain? Yahan sab samajhte hain!

🚗 Speed aur Acceleration ki Kahani

Ek car ka speedometer jo value dikha raha hai — woh speed function ki derivative hai position function ki. Aur agar speedometer ka reading suddenly jump kare (discontinuous ho) — toh physically kya hoga? Car ek jagah se doosri jagah teleport ho jaayegi — jo impossible hai! Isliye real physical functions continuous hote hain. Calculus mein hum formally define karte hain ki "continuous" ka kya matlab hai — aur phir differentiation explore karte hain.

1. Continuity at a Point

f(x) is continuous at x = a iff:

(i) f(a) defined ho (value exist kare)

(ii) lim[x→a] f(x) exist kare (left limit = right limit)

(iii) lim[x→a] f(x) = f(a) (limit = function value)

Types of Discontinuity ✅ Continuous ⚠️ Removable (Hole) ❌ Jump

2. Differentiability

f(x) differentiable at x=a iff: f'(a) = lim[h→0] [f(a+h)-f(a)]/h exist kare (finite ho).

Key Rule: Differentiable ⟹ Continuous. But Continuous ⟹ Differentiable NAHI! (Example: |x| at x=0 — continuous hai lekin differentiable nahi!)

3. Chain Rule

If y = f(g(x)), then: dy/dx = f'(g(x)) × g'(x) Example: y = sin(x²) dy/dx = cos(x²) × 2x = 2x·cos(x²) Nested: y = (3x²+1)⁵ dy/dx = 5(3x²+1)⁴ × 6x = 30x(3x²+1)⁴

4. Important Differentiation Formulas

d/dx(sin⁻¹x) = 1/√(1-x²) d/dx(cos⁻¹x) = -1/√(1-x²) d/dx(tan⁻¹x) = 1/(1+x²) d/dx(eˣ) = eˣ d/dx(ln x) = 1/x d/dx(aˣ) = aˣ ln(a) d/dx(xⁿ) = nxⁿ⁻¹

5. Logarithmic Differentiation

Jab function mein variable dono base aur exponent mein ho (jaise xˣ, xsin(x)) — tab log le lo pehle!

✅ Differentiate y = xˣ

Step 1: ln y = x ln x
Step 2: Differentiate both sides: (1/y)(dy/dx) = ln x + x·(1/x) = ln x + 1
Step 3: dy/dx = y(ln x + 1) = xˣ(ln x + 1)

6. Implicit Differentiation

✅ Find dy/dx if x² + y² = 25

Differentiate both sides w.r.t. x: 2x + 2y(dy/dx) = 0

dy/dx = -x/y

7. Second Order Derivative

y = f(x) y' = f'(x) = dy/dx (first derivative) y'' = f''(x) = d²y/dx² (second derivative = rate of rate of change) Physical meaning: y=displacement, y'=velocity, y''=acceleration

8. Practice Questions

Q1. Differentiate y = sin(e^(x²)) w.r.t. x.

Solution

Chain rule twice: dy/dx = cos(e^(x²)) × e^(x²) × 2x = 2x·e^(x²)·cos(e^(x²))

Q2. Find dy/dx if y = tan⁻¹(x/(1-x²)).

Solution

Hint: x/(1-x²) = ... use substitution x=tan θ. Answer: 2/(1+x²)

Q3. If y = (sin x)ˢⁱⁿˣ, find dy/dx.

Solution

ln y = sin x · ln(sin x). Differentiate: (1/y)dy/dx = cos x·ln(sin x) + sin x·(cos x/sin x) = cos x(ln(sin x)+1). dy/dx = (sin x)^(sin x)·cos x·(ln sin x + 1)