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A B

C D

Figure 3-1 At A and B, the dashed lines are tangent to the curves at the points shown by the dots. At C and D, the lines are not tangent to the curves.

(x₀,y₀) (x,y) y=f(x) Δy Δx Slope of line = Δy/Δx

Figure 3-2 The slope of a curve at a point (x0,y0) can be approximated by constructing a line through that point and a nearby point (x,y), and then finding the slope of the line, Δy/Δx.

and the difference in the x-values is

Δx=x−x0

The slope of the line through the two points is

Δy /Δx= ( y−y0) / (x−x0)

This slope is approximately the same as M. The accuracy of our approximation depends on how close together the two points are. We’ll get the best results if we choose (x,y ) as close to (x0,y0) as

possible. But we can’t make the points identical in an effort to find M exactly. If we do that, we getΔy= 0 and Δx= 0. Then when we try to calculate the slope, we get 0/0. That’s no help!

Converging points

In Chap. 2, we reviewed the theory of limits, which Newton and Leibniz used centuries ago to figure out instantaneous rates of change in the values of functions. Isaac Newton called these rates of change fluxions. We call them derivatives.

In the situation of Fig. 3-2, we can’t put (x,y ) directly on top of the point (x0,y0); if we do,

our problem reduces to nonsense. But we can move (x,y ) toward (x0,y0) until the two points

are arbitrarily close together. We get the points as close to each other as we can imagine—and then a little closer! We minimize Δx until it’s “too small to see.”

AsΔx shrinks to almost nothing, Δy does the same. Imagine the point (x,y ) getting to within a hair’s width, then a bacterium’s length, then a proton’s diameter of (x0,y0). As this occurs,

the slope of the line through the two points gets arbitrarily close to M, as shown in Fig. 3-3. The actual value of M is therefore equal to the limit of Δy/Δx as Δx approaches 0:

M = Lim x Δ →0 Δy /Δx Vanishing Increments 37 (x₀,y₀) (x,y) y=f(x) Slope of curve at point = dy/dx Δy Δx

Figure 3-3 As (x,y) approaches (x0,y0), the increments Δy and

Δx become smaller, and the line approaches the slope of the curve at the point (x0,y0). We call this slope

We now have a way to find the slope of a curve or the instantaneous rate of change in any function, as long as we can find the limit of Δy /Δx as Δx approaches 0. This limit is the derivative of the function.

Mathematicians call the arbitrarily tiny quantities Δy and Δx differentials, and sometimes symbolize them dy and dx,respectively. They are called infinitesimals in some texts. We can express the exact “mystery slope” as

M= dy /dx

Are you confused?

You might ask, “What’s the difference between y and f (x ) here? Are they different names for the same thing?” If y is the dependent variable in a function f (x ), then they are indeed the same. As you work with calculus in the future, you’ll likely see the derivative of a function y =f (x ) written in many different ways. Here are some of the variants you should watch for:

dy /dx y′ df(x ) / dx d / dx f (x ) df /dx f ′(x ) f ′ Here’s a challenge!

Find the slope M of a line tangent to the graph of the function

f (x ) =x2

at the point (x0,y0)= (1,1) in rectangular xy-coordinates, where y = f (x ). What does this slope

represent?

Solution

Let’s set up a two-point scheme, choosing a movable point (x,y ) near (1,1) as shown in Fig. 3-4. The func- tion tells us that y=x 2 _{for all possible values of x and y. The x-value of our movable point is}

x= 1 + Δx

They-value of our movable point is

y = 1 + Δy= (1 + Δx )2_{= 1 + 2Δx+ (Δx )}2

The slope of the line passing through our two points is

Substituting the quantity [1 + 2Δx+ (Δx )2_{] in place of y in the above equation, and also substituting the}

quantity (1 + Δx ) in place of x, we get

Δy /Δx= [1 + 2Δx+ (Δx )2_{− 1] / (1 + Δx− 1)}

which can be simplified to

Δy /Δx= [2Δx+ (Δx )2_{] / Δx}

and further to

Δy /Δx= 2 + Δx

To find the slope of the line tangent to the curve at the point (1,1), we must find

M= Lim x

Δ →0 Δy /Δx In this case, it turns out to be

M= Lim x

Δ →0 2 + Δx

It’s easy to see that the quantity (2 + Δx ) approaches 2 as Δx approaches 0. That limit, 2, is the slope of the line tangent to the curve at the point (1,1). It’s also the derivative of the function f (x ) =x 2 _{at the point}

where x = 1. 3 2 1 0 1 2 3 0 (1,1) f(x) = x2 x Δy Δx Slope of line = Δy/Δx yor f(x) Movable point (x,y) = (1+Δx, 1+Δy)

Figure 3-4 Finding the slope of a line tangent to the graph of

f (x)= x 2 _{at the point (1,1).}