Differentiation

The main objective is to define differentiability at a point. The derivative of a function is also given.  Techniques on differentiation include  product rule , quotient rule and Chain rule.
Two thereoms, namely the Mean value Theorem and  Rolle'sTheorem,   for functions which are  continuous on a closed and bounded interval [a,b] and differentiable on the open interval (a,b) will be  discussed.

Syllabus

• Differentiability and derivatives at points.
• Rules on differentiation: Sum rule, difference rule, quotient rule, product rule, and  chain rule.
• Differentiation of sine, cosine and the tangent functions.
• Implicit differentiation.
• Relative extrema; stationary and critical points.
• Absolute extrema.
• The Extreme Value Theorem.
• Rolle's Theorem and Mean Value Theorem.
• Increasing and decreasing functions.
• The derivative test for relative extrema.
• Concavity and point of inflection.
• Second derivative test for relative extrema.
• Application to graph sketching.
• Taylor's formula.
• l'Hôpital's rule.
• Some consequences or applications.

Reference: NG TB, Calculus, Chapter 5.