Calculus / Analysis CA-001

Fundamentals of Differential Calculus

An introduction to limits, derivatives, and the fundamental theorem of calculus, presented in the classical mathematical style.

Goals

In this note, we establish the foundational concepts of differential calculus:

  1. The formal definition of a limit
  2. The derivative as a limit of difference quotients
  3. Basic differentiation rules
  4. The connection between derivatives and tangent lines
Notation

Throughout this note, we adopt the following conventions:

  • denotes the set of real numbers
  • indicates a function from set to set
  • means the limit of as approaches equals
  • or denotes the derivative of with respect to

The Concept of a Limit

Definition: Limit of a Function

Let be a function defined on some open interval containing , except possibly at itself. We say that the limit of as approaches is , written

if for every there exists a such that

Remark

The - definition captures the intuitive notion that can be made arbitrarily close to by taking sufficiently close to (but not equal to) . This formalization, due to Weierstrass, provides the rigorous foundation for all of calculus.

The Derivative

Definition: Derivative at a Point

The derivative of a function at a point , denoted , is defined as

provided this limit exists. When it exists, we say is differentiable at .

Theorem: Power Rule

For any real number and function , we have

Proof

Using the definition of derivative:

Expanding using the binomial theorem:

Therefore:

Taking the limit as , all terms containing vanish, leaving .

Differentiation Rules

Theorem: Sum and Product Rules

Let and be differentiable functions. Then:

Sum Rule:

Product Rule:

Example

Consider . By the sum rule and power rule:

For , by the product rule:

The Chain Rule

Theorem: Chain Rule

If is differentiable at and is differentiable at , then the composite function is differentiable at , and

In Leibniz notation, if and , then

Remark

The chain rule is perhaps the most important differentiation technique, as it allows us to differentiate composite functions. It also provides the foundation for implicit differentiation and related rates problems.

Further Reading

These notes cover only the basics. For a rigorous treatment, consult:

  • Rudin, W. Principles of Mathematical Analysis (1976)
  • Spivak, M. Calculus (2008)
  • Apostol, T. Calculus, Vol. 1 (1967)