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📉 Limits and Derivatives

1. Limits

The concept of limit is fundamental to calculus. Intuitively, lim (x→a) f(x) = L means that as x gets very close to 'a', f(x) gets very close to 'L'.

Standard Limits

1. lim (x→a) (xⁿ - aⁿ) / (x - a) = n aⁿ⁻¹

2. lim (x→0) (sin x) / x = 1

3. lim (x→0) (tan x) / x = 1

4. lim (x→0) (e^x - 1) / x = 1

2. Derivatives (First Principle)

The derivative of a function f at x is defined as the rate of change of the function. Geometrically, it is the slope of the tangent to the curve.

f'(x) = lim (h→0) [f(x+h) - f(x)] / h First Principle of Differentiation

3. Rules of Differentiation

  • Sum/Difference Rule: d/dx [u ± v] = du/dx ± dv/dx
  • Product Rule: d/dx [uv] = u(dv/dx) + v(du/dx)
  • Quotient Rule: d/dx [u/v] = [v(du/dx) - u(dv/dx)] / v²

4. Standard Derivatives

Basic Formulas

d/dx (xⁿ) = n xⁿ⁻¹

d/dx (sin x) = cos x

d/dx (cos x) = -sin x

d/dx (e^x) = e^x