Derivatives
A speedometer reads a speed at a single instant. How is a rate of change at one point defined, and how can it be found for any formula without taking a limit every time?
Where you are 13 inside
What this area is about
The derivative f'(a) is the limit of secant slopes, the slope of a curve at one point and the instantaneous rate of change there. Computing that limit by hand every time would be slow, so most of this area builds rules that give derivatives of whole families of functions directly.
How these fit together
Definition and meaning. Definition of the derivative, an area of its own, defines f' as a limit. The derivative as a rate of change reads its units and sign. Differentiability shows where a continuous graph still has no derivative.
Rules. The power, constant multiple and sum rules handle polynomials; the product rule, the quotient rule and the chain rule handle products, quotients and compositions.
Families. Trig, exponential and log functions (an area), inverse functions and inverse trig functions.
Techniques. Second and higher derivatives differentiate again. Implicit differentiation finds slopes on curves not solved for y. Combining rules decides the order for layered expressions.
A path through it
The definition first; the power rule, the rate interpretation and differentiability each build on it. Then the product, quotient and chain rules, in that order. Next the exponential and trig derivatives, then second derivatives, then implicit differentiation, which also uses tangent lines; the logarithm lesson comes after it. Inverse functions, then inverse trig. Combining rules draws on all of them, so it comes last.
Next come applications of derivatives and antiderivatives.