Learning Objectives
- Understand continuity of functions and its conditions
- Learn differentiability and its relation to continuity
- Study chain rule, implicit differentiation, parametric differentiation
- Learn logarithmic differentiation and higher-order derivatives
- Understand Rolle's theorem and Mean Value Theorem
Key Concepts
Continuity
A function f is continuous at x = a if: (1) f(a) is defined, (2) limx→a f(x) exists (LHL = RHL), (3) limx→a f(x) = f(a). A function is continuous on an interval if continuous at every point. Polynomial, exponential, sine, cosine functions are continuous everywhere. |x| is continuous everywhere but not differentiable at x = 0.
Algebra of continuous functions: Sum, difference, product, quotient (denominator ≠ 0), and composition of continuous functions are continuous.
Differentiability
f is differentiable at x = a if f'(a) = limh→0 [f(a+h) - f(a)]/h exists (LHD = RHD). Differentiability implies continuity (but not converse). Example: |x| is continuous at 0 but not differentiable (sharp corner).
Differentiation Rules
Chain Rule: If y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). Product Rule: (uv)' = u'v + uv'. Quotient Rule: (u/v)' = (u'v - uv')/v2.
Implicit differentiation: For equations like x2 + y2 = r2, differentiate both sides with respect to x, treating y as a function of x. Parametric differentiation: If x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt) = g'(t)/f'(t).
Logarithmic differentiation: Take natural log of both sides; useful for y = [f(x)]g(x) or products of many functions. Example: y = xx → ln y = x ln x → y'/y = 1 + ln x → y' = xx(1 + ln x).
Standard Derivatives
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.
Higher Order Derivatives
Second derivative: d2y/dx2 = d/dx(dy/dx). For parametric: d2y/dx2 = [d/dt(dy/dx)] / (dx/dt).
Mean Value Theorems
Rolle's Theorem: If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists c ∈ (a,b) such that f'(c) = 0.
Lagrange's Mean Value Theorem (LMVT): If f is continuous on [a,b] and differentiable on (a,b), then there exists c ∈ (a,b) such that f'(c) = [f(b) - f(a)]/(b - a). Geometrically: tangent at c is parallel to secant joining (a,f(a)) and (b,f(b)).
Summary
Continuity requires the function value to equal the limit. Differentiability implies continuity. Chain rule, implicit differentiation, and logarithmic differentiation extend the basic rules. Rolle's theorem and LMVT connect function values to derivative values.
Important Terms
- Continuous: No breaks, jumps, or holes in the graph
- Differentiable: Derivative exists; implies continuity
- Chain rule: dy/dx = (dy/du)(du/dx) for composite functions
- LMVT: f'(c) = [f(b)-f(a)]/(b-a) for some c in (a,b)
- Rolle's theorem: Special case of LMVT when f(a) = f(b)
Quick Revision
- Continuous at a: lim f(x) = f(a) as x→a
- Differentiable ⟹ Continuous (converse false: |x| at 0)
- Chain rule: (fog)' = f'(g(x)) · g'(x)
- d/dx(sin-1x) = 1/√(1-x2); d/dx(tan-1x) = 1/(1+x2)
- Logarithmic: for f(x)g(x), take ln both sides
- Parametric: dy/dx = (dy/dt)/(dx/dt)
- LMVT: f'(c) = [f(b)-f(a)]/(b-a)