Introduction to Limits & Derivatives

This module opens the door to the differential calculus by developing the limit concept carefully and then using it to define and compute derivatives. Learners examine average and instantaneous rates of change, tangent lines and the standard rules of differentiation. The physical interpretation of the derivative is given equal weight with the symbolic one. By its end, learners can differentiate a wide class of elementary functions and interpret the result in context.

Learning Outcomes

Evaluate limits of elementary functions, including one-sided and infinite limits (apply)

Identify and classify discontinuities of a given function (analyse)

Compute derivatives from first principles for simple functions (apply)

Apply the power, product, quotient and chain rules to differentiate standard functions (apply)

Determine equations of tangent and normal lines at a given point (apply)

Interpret a derivative as a rate of change in a physical setting (evaluate)

Construct a model of a rate-of-change problem and solve it (create)

Analyse the connection between continuity, differentiability and the derivative (analyse)

Topics Covered

Informal limits and the limit notation

Limit laws and algebraic evaluation

One-sided and infinite limits

Limits at infinity and horizontal asymptotes

Continuity and types of discontinuity

The derivative as a limit

Derivative from first principles

Power rule and polynomial derivatives

Product and quotient rules

Chain rule and composite functions

Derivatives of exponential and logarithmic functions

Derivatives of trigonometric functions

Tangent and normal lines

Velocity, acceleration and physical rates

Introduction to higher derivatives

Skills Developed

Evaluating limits algebraically Classifying discontinuities Differentiating from first principles Applying the standard differentiation rules Computing tangent and normal equations Translating word problems into derivative form Interpreting derivatives physically Checking a derivative against its graph

Suggested Resources

STRAIT Introduction to Calculus workbook (TutorDA, 2025)

Stewart, J. Calculus (8th ed.), chapters 1-3

Khan Academy: Differential Calculus playlist

3Blue1Brown: Essence of calculus

Paul's Online Math Notes: Calculus I

Desmos derivative explorer

TutorDA practice bank: A-M4 drill sets

Assessment

Assessment combines a written examination that tests computational facility with coursework in which learners write up extended problems linking derivatives to real-world rates of change.

Written examination (2 hr)60%

Coursework and problem write-ups40%

Practice bank: 199 MCQs · Timed mock & self-paced modes

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