HSSLIVE Plus One Maths Chapter 13: Limits and Derivatives Notes

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Limits and derivatives form the cornerstone of calculus, introducing students to the mathematics of change and motion. This chapter covers the intuitive and formal approaches to limits, continuity of functions, and the derivative as a rate of change. Students will learn differentiation techniques and applications including slope of curves, tangent lines, rates of change, and optimization problems. These concepts provide essential tools for modeling dynamic systems in physics, economics, biology, and numerous other fields.

Chapter 13: Limits and Derivatives

Limits:

The limit of a function f(x) as x approaches a value c is the value that f(x) gets arbitrarily close to as x gets arbitrarily close to c.

  • Notation: lim(x→c) f(x) = L

Properties of Limits:

  1. lim(x→c) [f(x) ± g(x)] = lim(x→c) f(x) ± lim(x→c) g(x)
  2. lim(x→c) [f(x) × g(x)] = lim(x→c) f(x) × lim(x→c) g(x)
  3. lim(x→c) [f(x)/g(x)] = lim(x→c) f(x)/lim(x→c) g(x), if lim(x→c) g(x) ≠ 0

Evaluating Limits:

  • Direct substitution
  • Factoring
  • Rationalization
  • L’Hôpital’s rule (for indeterminate forms)

Important Limits:

  1. lim(x→0) (sin x)/x = 1
  2. lim(x→0) (e^x – 1)/x = 1
  3. lim(x→0) (a^x – 1)/x = ln a

Derivatives:

The derivative of a function f(x) with respect to x is the instantaneous rate of change of f(x) with respect to x.

  • Notation: f'(x) or df/dx
  • Definition: f'(x) = lim(h→0) [f(x+h) – f(x)]/h

Basic Differentiation Rules:

  1. d/dx(c) = 0 (where c is a constant)
  2. d/dx(x^n) = nx^(n-1)
  3. d/dx(e^x) = e^x
  4. d/dx(ln x) = 1/x
  5. d/dx(sin x) = cos x
  6. d/dx(cos x) = -sin x
  7. d/dx(tan x) = sec²x

Laws of Differentiation:

  1. d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
  2. d/dx[f(x) × g(x)] = f'(x)g(x) + f(x)g'(x) (Product Rule)
  3. d/dx[f(x)/g(x)] = [f'(x)g(x) – f(x)g'(x)]/[g(x)]² (Quotient Rule)
  4. d/dx[f(g(x))] = f'(g(x)) × g'(x) (Chain Rule)

Applications of Derivatives:

  • Rate of change
  • Tangent and normal to curves
  • Increasing/decreasing functions
  • Maxima and minima
  • Approximations

 

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