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Differential Calculus

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Limits and Continuity

Key Concepts:

Limit Definition: lim(x→a) f(x) = L
Left Limit: lim(x→a⁻) f(x)
Right Limit: lim(x→a⁺) f(x)
Continuity: f is continuous at x=a if lim(x→a) f(x) = f(a)

Standard Limits:

lim(x→0) sin(x)/x = 1

lim(x→0) (1-cos(x))/x² = 1/2

lim(x→0) (eˣ-1)/x = 1

lim(x→0) ln(1+x)/x = 1

lim(x→0) (aˣ-1)/x = ln(a)

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Differentiation Rules

Basic Rules:

• d/dx(c) = 0
• d/dx(x) = 1
• d/dx(xⁿ) = nx^(n-1)
• d/dx(cf) = c·f'
• d/dx(f±g) = f'±g'

Product & Quotient:

• (fg)' = f'g + fg'
• (f/g)' = (f'g - fg')/g²
• Chain Rule: (f∘g)' = f'(g)·g'

Special Functions:

• d/dx(eˣ) = eˣ
• d/dx(ln x) = 1/x
• d/dx(aˣ) = aˣ ln(a)
• d/dx(logₐ x) = 1/(x ln a)

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Trigonometric Derivatives

Standard Derivatives:

d/dx(sin x) = cos x

d/dx(cos x) = -sin x

d/dx(tan x) = sec²x

d/dx(cot x) = -csc²x

d/dx(sec x) = sec x tan x

d/dx(csc x) = -csc x cot x

Inverse Trigonometric:

d/dx(sin⁻¹x) = 1/√(1-x²)

d/dx(cos⁻¹x) = -1/√(1-x²)

d/dx(tan⁻¹x) = 1/(1+x²)

d/dx(cot⁻¹x) = -1/(1+x²)

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Applications of Derivatives

Rate of Change:

• Instantaneous Rate: f'(x) = lim(h→0) [f(x+h)-f(x)]/h
• Related Rates: dx/dt, dy/dt relationships
• Velocity: v = ds/dt
• Acceleration: a = dv/dt = d²s/dt²

Tangent & Normal:

• Slope of tangent at (a,f(a)): m = f'(a)
• Tangent: y - f(a) = f'(a)(x - a)
• Normal: y - f(a) = -1/f'(a)(x - a)

Maxima & Minima:

• Critical Points: f'(x) = 0 or f'(x) undefined
• First Derivative Test: f'(x) changes sign
• Second Derivative Test: f''(x) > 0 (min), f''(x) < 0 (max)
• Global extrema on [a,b]: check endpoints

Curve Sketching:

• Increasing: f'(x) > 0
• Decreasing: f'(x) < 0
• Concave Up: f''(x) > 0
• Concave Down: f''(x) < 0
• Inflection Point: f''(x) = 0 and changes sign

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Important Theorems

L'Hôpital's Rule:

For 0/0 or ∞/∞ forms:
lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x)

Conditions: f(a) = g(a) = 0 or both → ∞, and g'(x) ≠ 0 near x = a

Mean Value Theorem:

If f is continuous on [a,b] and differentiable on (a,b):
∃c ∈ (a,b): f'(c) = [f(b)-f(a)]/(b-a)

Rolle's Theorem: If f(a) = f(b), then ∃c: f'(c) = 0

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Higher Order Derivatives

Notation:

• f'(x) = dy/dx
• f''(x) = d²y/dx²
• f'''(x) = d³y/dx³
• f⁽ⁿ⁾(x) = dⁿy/dxⁿ

Leibniz Rule:

(fg)⁽ⁿ⁾ = Σ C(n,k) f⁽ᵏ⁾g⁽ⁿ⁻ᵏ⁾

where C(n,k) are binomial coefficients

Applications:

• Concavity Analysis
• Taylor Series
• Optimization Problems
• Physics Applications

Quick Formula Reference

Basic Derivatives

d/dx(xⁿ) = nx^(n-1)

d/dx(eˣ) = eˣ

d/dx(ln x) = 1/x

Chain Rule

d/dx[f(g(x))] = f'(g(x))·g'(x)

(uv)' = u'v + uv'

Trigonometric

d/dx(sin x) = cos x

d/dx(tan x) = sec²x

Applications

Critical points: f'(x) = 0

Inflection: f''(x) = 0

L'Hôpital

0/0: lim f/g = lim f'/g'

∞/∞: lim f/g = lim f'/g'

Higher Order

f''(x) = d²y/dx²

Taylor: f(x) ≈ Σ f⁽ⁿ⁾(a)/n!

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