Ch 5 covers continuity, differentiability, and advanced differentiation techniques: chain rule, implicit, logarithmic, parametric, and second-order derivatives. It also introduces Rolle's and Mean Value Theorems.
A function f is continuous at x=a if: (1) f(a) is defined, (2) lim(x→a) f(x) exists, (3) lim(x→a) f(x) = f(a). All polynomial, trig, exponential, and log functions are continuous in their domains.
Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x). Implicit: differentiate both sides and solve for dy/dx. Logarithmic: take log first, then differentiate. Parametric: dy/dx = (dy/dt)/(dx/dt).
Rolle's: if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃ c ∈ (a,b) such that f'(c)=0. MVT: ∃ c ∈ (a,b) such that f'(c) = [f(b)−f(a)]/(b−a).
Download: https://ncert.nic.in/textbook/pdf/lemh105.pdf | Complete book Part I: https://ncert.nic.in/textbook/pdf/lemh1ps.zip
Every differentiable function is continuous, but not every continuous function is differentiable. Example: f(x) = |x| is continuous at x=0 but not differentiable there (sharp corner).
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