Lecture Notes on Basic Math: Differentiation and Integration

Introduction

• Lecturer: Rohit Gupta
• Topic: Differentiation and Integration in Basic Math
• Objective: Eliminate fear of these topics.

Importance

• Part of mathematics and essential for science, especially physics.
• Two parts of the course:
  1. How to differentiate and integrate.
  2. Understanding the meaning and applications of differentiation and integration.

Differentiation

Basics

• Example: Dividing numbers to find a quotient (e.g., 27 divided by 3 equals 9).
• Differentiation is about understanding rates of change between variables.
• Symbols: Commonly written as dy/dx.
• Purpose: To find the rate of change of a function.
• Variables: Involves independent and dependent variables related by a function.

Types of Functions

• Constant Functions: No change in value.
• Polynomial Functions: Expressions like x^n (e.g., x squared).
• Trigonometric Functions: Sine, cosine, etc.
• Logarithmic Functions: Involves log expressions.
• Exponential Functions: Involves expressions like e^x.

Rules and Examples

• Power Rule: The derivative of x^n is n*x^(n-1).
• Trig Functions: Differentiation rules for sine, cosine, etc., (e.g., derivative of sin(x) is cos(x)).
• Logs and Exponential: Differentiation of ln(x) is 1/x, e^x is e^x.
• Product Rule: For u*v, the derivative is u'*v + u*v'.
• Quotient Rule: For u/v, the derivative is (u'*v - u*v')/v^2.

Integration

Basics

• Definition: Reverse process of differentiation, involves summing area under a curve.
• Two Types: Definite (with limits) and indefinite (without limits).
• Symbols: Integral sign (∫).

Key Concepts

• Definite Integrals: Result in a numerical value as they are evaluated over an interval [a, b].
• Indefinite Integrals: Result in a family of functions representing an antiderivative plus a constant C.
• Fundamental Theorem of Calculus: Links differentiation and integration.

Examples and Rules

• Constant Function Integration: Integral of c is c*x + C.
• Power Rule: Integral of x^n is (x^(n+1))/(n+1) + C, for n ≠ -1.
• Basic Trigonometric Functions: Integrals
  • Integral of sin(x) is -cos(x) + C.
  • Integral of cos(x) is sin(x) + C.
• Logarithmic Functions: Integral of 1/x is ln|x| + C.
• Exponential Functions: Integral of e^x is e^x + C.

Practical Tips & Practice Problems

• Practice differentiating and integrating various functions.
• Utilize fundamental rules and memorize key differentiation and integration formulas.
• *Problem Examples:
  1. Differentiate x^3.
  2. Integrate 2x^2.
  3. Differentiate sin(x) * ln(x) (Product Rule).
  4. Integrate 1/x from 2 to 5 (Definite Integral).

Conclusion

• Differentiation and integration are essential tools in mathematics, especially for physics.
• Practice is necessary to achieve fluency.
• Review and understand the meaning behind the operations, not just the procedures.