Differential Calculus Overview

Overview

This lecture covers the main concepts and rules of differential calculus, including limits, differentiation techniques, applications, stationary points, and the connection between differentiation and integration.

Properties of Limits

• The limit of a sum is the sum of the limits, e.g., limₓ→ₐ[f(x) + g(x)] = limₓ→ₐf(x) + limₓ→ₐg(x).
• The limit of a quotient is the quotient of the limits, provided the denominator is not zero.
• Constants can be factored out of limits: limₓ→ₐ[c·g(x)] = c·limₓ→ₐg(x).
• Limits involving roots: limₓ→ₐ[√(f(x) - g(x))] = [limₓ→ₐf(x) - limₓ→ₐg(x)]¹/².

Finding Limits of Functions

• Substitute the value of x directly unless it yields an indeterminate form (e.g., 0/0).
• Use L'Hospital's Rule for indeterminate forms: differentiate numerator and denominator, then substitute.
• Limits that result in division by zero are undefined; the limit does not exist.

Rules and Examples of Differentiation

• Power Rule: d/dx[xⁿ] = n·xⁿ⁻¹.
• The derivative of a constant is zero.
• Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x).
• Product Rule: d/dx[u·v] = u'·v + u·v'.
• Quotient Rule: d/dx[u/v] = (u'·v - u·v')/v².
• The derivative of ln(x) is 1/x.
• The derivative of eˣ is eˣ.
• The derivative of sin(x) is cos(x); of cos(x) is -sin(x).

Differentiation Using Chain Rule

• Substitute an inner function as u, differentiate y with respect to u, then multiply by du/dx.
• Useful for composite functions like (x² + 2x + 1)⁶.

Differentiation of Parametric Equations

• For equations where x and y are functions of t: dy/dx = (dy/dt) / (dx/dt).
• The second derivative d²y/dx² = d/dx(dy/dx).

Applications of Differentiation

• To find maximum and minimum values (stationary points), set the first derivative to zero.
• Use the second derivative to determine whether a stationary point is a maximum (negative) or a minimum (positive).
• Application in real-world problems, e.g., maximizing dosage for blood pressure.

Stationary Points and Curve Analysis

• Stationary points occur where dy/dx = 0.
• Find y-coordinates by substituting x-values into the original function.
• Classify points using the second derivative: maxima, minima, or saddle points.

Relationship Between Differentiation and Integration

• Differentiation gives the rate of change; integration reverses differentiation to find the original function.
• If derivative of f(x) is g(x), then ∫g(x)dx = f(x) + C.
• Example: d/dx[ln(x²+1)] = 2x/(x²+1); ∫2x/(x²+1)dx = ln(x²+1) + C.

Key Terms & Definitions

• Limit — The value a function approaches as x approaches a certain point.
• Derivative — The instantaneous rate of change of a function with respect to its variable.
• Chain Rule — Method for differentiating composite functions.
• Product Rule — Rule for differentiating a product of two functions.
• Quotient Rule — Rule for differentiating the quotient of two functions.
• Stationary Point — Point where the first derivative is zero.
• Maximum/Minimum — Points where the function reaches highest/lowest value locally.
• Saddle Point — Stationary point that is neither maximum nor minimum.
• L'Hospital's Rule — Technique for evaluating indeterminate limits.

Action Items / Next Steps

• Review homework on limit calculations and differentiation rules.
• Practice finding stationary points and classifying them using the second derivative.
• Read about the relationship between differentiation and integration.