المشتقة

DIFFERENTIATION
OVERVIEW In the beginning of Chapter 2 we discussed how to determine the slope of a
curve at a point and how to measure the rate at which a function changes. Now that we
have studied limits, we can define these ideas precisely and see that both are interpretations
of the derivative of a function at a point. We then extend this concept from a single
point to the derivative function, and we develop rules for finding this derivative function
easily, without having to calculate any limits directly. These rules are used to find derivatives
of most of the common functions reviewed in Chapter 1, as well as various combinations
of them. The derivative is one of the key ideas in calculus, and we use it to solve a
wide range of problems involving tangents and rates of change
Tangents and the Derivative at a Point
In this section we define the slope and tangent to a curve at a point, and the derivative
of a function at a point. Later in the chapter we interpret the derivative as the instantaneous
rate of change of a function, and apply this interpretation to the study of certain
types of motion.
Finding a Tangent to the Graph of a Function
To find a tangent to an arbitrary curve at a point we use the procedure
introduced in Section 2.1. We calculate the slope of the secant through P and a nearby point
We then investigate the limit of the slope as (Figure 3.1). If the
limit exists, we call it the slope of the curve at P and define the tangent at P to be the line
through P having this slope.