| --- |
| source: ncert |
| topic: calculus |
| class_level: class_12 |
| chapter: ch5 |
| difficulty: advanced |
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| |
| Class 12 | Ch5: Continuity and Differentiability |
| CONTINUITY at x=a: lim(x→a) f(x) = f(a). |
| Three conditions: f(a) defined, limit exists, limit equals f(a). |
| Continuous function: continuous at every point in domain. |
| Sum, difference, product of continuous functions is continuous. |
| Composite of continuous functions is continuous. |
| DIFFERENTIABILITY: f'(a)=lim(h→0) [f(a+h)-f(a)]/h exists. |
| Differentiable → Continuous. Continuous does NOT imply differentiable. |
| |x| is continuous everywhere but NOT differentiable at x=0. |
| CHAIN RULE: d/dx[f(g(x))]=f'(g(x))×g'(x). dy/dx=(dy/du)×(du/dx). |
| IMPLICIT DIFFERENTIATION: differentiate both sides w.r.t. x, treat y as function of x. |
| d/dx[f(y)]=f'(y)×dy/dx. |
| PARAMETRIC DIFFERENTIATION: x=f(t), y=g(t). dy/dx=(dy/dt)/(dx/dt)=g'(t)/f'(t). |
| LOGARITHMIC DIFFERENTIATION: for y=x^x or y=f(x)^g(x). Take log both sides, differentiate. |
| SECOND ORDER DERIVATIVE: d^2y/dx^2=d/dx[dy/dx]. |
| STANDARD DERIVATIVES: |
| d/dx[x^n]=nx^(n-1). d/dx[e^x]=e^x. d/dx[a^x]=a^x ln(a). |
| d/dx[ln x]=1/x. d/dx[log_a x]=1/(x ln a). |
| d/dx[sin x]=cos x. d/dx[cos x]=-sin x. d/dx[tan x]=sec^2 x. |
| d/dx[sin^(-1)x]=1/sqrt(1-x^2). d/dx[cos^(-1)x]=-1/sqrt(1-x^2). |
| d/dx[tan^(-1)x]=1/(1+x^2). |
| ROLLE'S THEOREM: if f continuous on [a,b], differentiable on (a,b), f(a)=f(b), then there exists c in (a,b) where f'(c)=0. |
| MEAN VALUE THEOREM (MVT/Lagrange): if f continuous on [a,b], differentiable on (a,b), then there exists c where f'(c)=[f(b)-f(a)]/(b-a). |
| SOLVED EXAMPLES: |
| Example 1: Differentiate x^x. |
| y=x^x → ln y=x ln x → (1/y)(dy/dx)=ln x+1 → dy/dx=x^x(1+ln x). |
| Example 2: If x=at^2, y=2at, find dy/dx. |
| dx/dt=2at, dy/dt=2a. dy/dx=2a/2at=1/t. |
| Example 3: Check continuity of |x-3| at x=3. |
| LHL=lim(x→3-)|x-3|=lim(x→3-)(3-x)=0. RHL=lim(x→3+)(x-3)=0. f(3)=0. Continuous. |
| But not differentiable at x=3 (left derivative=-1, right=+1, unequal). |
| COMMON MISTAKES: |
| Continuous does NOT mean differentiable. |x| is the standard counter-example. |
| Chain rule: derivative of OUTER function × derivative of INNER function. |
| In parametric: dy/dx=(dy/dt)/(dx/dt), NOT (dx/dt)/(dy/dt). |
