Continuity, Differentiability and Applications of Derivatives

About this unit

This unit moves from the idea of a limit to the two properties that make calculus work: continuity (no breaks in a graph) and differentiability (a well-defined slope at every point). You then learn rules for differentiating composite, inverse trigonometric, implicit, exponential, logarithmic and parametric functions. The second half applies derivatives to real questions: how fast a quantity changes, where a function rises or falls, and where it reaches its highest or lowest value. These skills feed directly into integration, differential equations and applied calculus.

Key ideas

• f is continuous at c when the limit of f(x) as x → c exists and equals f(c).
• Every differentiable function is continuous, but a continuous function need not be differentiable.
• The chain rule: d/dx f(g(x)) = f′(g(x)) · g′(x).
• Logarithmic, implicit and parametric differentiation handle awkward functions.
• dy/dx is the rate of change of y with respect to x.
• The sign of f′(x) tells whether f is increasing or decreasing.
• First and second derivative tests locate local maxima and minima.

How to prepare

Memorise the standard derivatives, especially those of the inverse trigonometric functions, eˣ and log x. Practise one-sided limits for piecewise functions. For applied problems, always write the quantity to be optimised as a function of one variable before differentiating, and check end points on closed intervals.