Calculus: Analytic Geometry and Calculus, with Vectors

168 Functions, limits, derivatives

to obtain

TXdlog lxi = d log (-x) -TX lx d(dxx) x

Thus we can extend the two formulas in (3.675) to obtain the more

general formulas

(3.676)

d

log lxI =

1X d

log Jul =

1 du

TX x TX u dx

in which it is required that x ; 0 and u 0 0 but it is not required that x

and u be positive.

Up to the present time, our work with difference quotients and deriva-

tives has involved only fundamental definitions and formulas. Figures

and geometric ideas, which might be helpful but which might also be mis-

Figure 3.68

leading, have been completely ab-

sent. Section 5.1 will present our

thorough introduction to matters

relating to slopes of graphs and

tangents to graphs. Meanwhile, we

may be helped and may be unmisled

by looking at Figure 3.68, which

shows the graph C of a differentiable

function f. The points P and Q hav-

ing coordinates (x,y) or (x, f(x)) and

(x + Ax, y + Ay) or (x + Ox, f(x + Ox)) are shown, but the line PQ

joining P and Q is not drawn. The first of the two formulas

(3.681)

y=f(x + Ox) - AX)= slope of line PQ

AX Ax

(3.682) dz = f'(x) = slope of tangent to C at P

= slope of C at P

is correct because it is obtained by applying the definition of the slope of a

line. The second formula is correct by definition; the line through P

whose slope is the limit as Ax approaches zero of the slopes in (3.681) is,

by definition, the line tangent to C at P, and, moreover, the slope of C at

P is, by definition, the slope of the line tangent to C at P. The definition

gives precision to the venerable idea that the slope of the line PQ is close

to the slope of the tangent at P whenever Q is close to P. The definition

(3.682) turns out to be very important. Indeed, there are many situ-

ations in which magnitudes play minor roles and it is important to know

that the graph C of y = f(x) has a horizontal tangent (tangent of zero

slope) at each point (x,y) on C for which f'(x) = 0, has a tangent of

positive slope at each point (x,y) on C for which f'(x) > 0, and has a

tangent of negative slope at each point (x,y) on C for which f' (x) < 0.