ABSTRACT
This chapter suggests ways that (parts of) a first course in differential calculus (differentiation, the derivative and related topics/concepts) can be taught and learnt meaningfully. There is truth in the old adage that everything has its pros and cons in approaches to differentiation. The rough and ready approach allows teachers and students to get straight into the ideas of pointwise differentiation without preparatory work on limits, infinitesimals or kinematics. The graph displays the curve for the given gradient function that was traced through negative coordinates, zero, and positive coordinates. The language, notations and representations teachers use to introduce the rules to students are interrelated. There are two standard ways to introduce the trigonometric ratios: via the ratios of sides of a right-angled triangle; via the unit circle. It focuses on higher derivatives, increasing and decreasing functions and the shape of functions/curves. Setting appropriate tasks is of course, essential for student learning.
