Calculus: Differentiation

Differentiation

Limits

• Definition: The limit of a function f(x) as x approaches a is denoted by lim x→a f(x) = L, if for every ε > 0, there exists a δ > 0 such that for all x, 0 < |x-a| < δ implies |f(x) - L| < ε.
• Properties:
  • Linearity: lim x→a [af(x) + bg(x)] = a lim x→a f(x) + b lim x→a g(x)
  • Homogeneity: lim x→a f(x) = f(a) if f is continuous at a
  • Squeeze Theorem: if f(x) ≤ g(x) ≤ h(x) and lim x→a f(x) = lim x→a h(x) = L, then lim x→a g(x) = L

Rules

• Power Rule: If f(x) = x^n, then f'(x) = nx^(n-1)
• Product Rule: If f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
• Quotient Rule: If f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
• Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Higher-Order Derivatives

• Definition: The nth derivative of a function f(x) is denoted by f^(n)(x) and is obtained by differentiating the (n-1)th derivative of f(x).
• Notation: f^(n)(x) = d^n/dx^n f(x) = d/dx (d^(n-1)/dx^(n-1) f(x))
• Properties:
  • Linearity: f^(n)(x) = af^(n)(x) + bf^(n)(x) if f(x) = af(x) + bf(x)
  • Leibniz's Formula: f^(n)(x) = ∑ (n choose k) * f^(k)(x) * g^(n-k)(x) if f(x) = f(x) * g(x)