The Slope of the Tangent and the Derivative

The previous section gave a rather vague definition of the tangent to a curve at a point, because we said that it is a straight line that just touches the curve at that point. We now give a more formal definition of the same concept.

The geometrical idea behind the definition is easy to understand. Consider a point P on a curve in the xy-plane (see Fig. 4.3). Take another point Q on the curve. The entire straight line through P and Q is called a secant (from a Latin word meaning "cutting"). If we keep P fixed, but let Q move along the curve toward P, then the secant will rotate around P, as indicated in Fig. 4.4. The limiting straight line PT toward which the secant tends is called the tangent (line) to the curve at P. Suppose that P is a point on the graph of the function /. We shall see how the preceding considerations enable us to find the slope of the tangent at P. This is shown in Fig. 4.5.

Point P has the coordinates (a, f(a)). Point Q lies close to P and is also on the graph of /. Suppose that the x-coordinate of Q is a + h, where h is a small number # 0. Then the x-coordinate of Q is not a (because Q ^ P), but a

FIGURE 4.3

FIGURE 4.4

FIGURE 4.5

number close to a. Because Q lies on the graph of /, the >-coordinate of Q is equal to f(a + h). Hence, the coordinates of the points are P = (a, f(a)) and Q = (a + h, f(a + h)). The slope mPQ of the secant PQ is mPQ =

In mathematics, this fraction is often called a Newton (or differential) quotient of /. Note that when h = 0, the fraction in [4.2] becomes 0/0 and so is undefined. But choosing h = 0 corresponds to letting Q = P. When Q moves toward P (Q tends to P) along the graph of /, the a:-coordinate of Q, which is a + h, must tend to a, and so h tends to 0. Simultaneously, the secant PQ tends to the tangent to the graph at P. This suggests that we ought to define the slope of the tangent at P as the number that mpQ in [4.2] approaches as h tends to 0. In [4.1], we called this value the slope f (a). So we propose the following definition of /'(a):

f(a\ = /the limit as f(a + h) 1 \ tends to 0 of J h m

In mathematics, it is common to use the abbreviated notation lim>,_o for "the limit as h tends to zero" of an expression involving h. We therefore have the following definition:

The derivative /'(a) of the function / at point a of its domain is given by the formula m = fin, /» + »>-/<«>

As in [4.1], the number f'(a) gives the slope of the tangent to the curve y = f(x) at the point (a, f(a)). The equation for a straight line passing through (*i,yO and having a slope b is given by y — yi = b(x — x\). Hence, we obtain:

The equation for the tangent to the graph of >• = /(*) at the point [a. f (a)) is y-f(a) = f'(a)(x-a) [4.4]

So far the concept of a limit in the definition of f'(a) is not quite clear. Section 6.7 gives a precise definition. Because it is relatively complicated, we rely on intuition for the time being. Consider a simple example.

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