The Calculus Primer (2011)
Part II. The Derivative of a Function
Chapter 6. THE MEANING OF THE DERIVATIVE
2—3. The Derivative as a Limiting Value. From what has been said above, the distinction between an average rate of change in a function, or , and the instantaneous rate of change, or the limiting value of , should begin to be clear. This limiting value of , the value, that is, of at a particular instant, is called a derivative. We represent this limiting value by the symbol . Thus,
The reader is urged, at this point, to note carefully that the symbols Δy and Δx each represent a definite quantity; hence the fraction represents the ratio of two quantities. But the symbol does not represent the ratio of two quantities; it should distinctly be regarded as a single symbol—as one quantity, the limiting value of an average rate. The symbols dy and dx have (as yet) no meaning as separate symbols.
2—4. Analytical Representation of the Derivative. Let us examine the derivative a little further. We take any point P(x1,y1) on the curve y = x2, and Q (x2,y2), another point on the curve, nearby. By making Δx, or x2 − x1, small enough, we can make the increment Δy, or y2 − y1, as small as we please. In short, as Δx → 0, Δy also approaches zero.
Now, it is easy to show that the average rate of change of y during the interval from x1 to x2 is given by
= 2x + Δx.
For,y = x2.(1)
y + Δy = (x + Δx)2,
ory + Δy = x2 + 2x·Δx + (Δx)2.(2)
Subtracting (1) from (2):
Δy = 2x·Δx+ (Δx)2.(3)
Dividing (3) by Δx:
= 2x + Δx.(4)
We may now let Δx approach zero as a limit; the average rate for the interval Δx then becomes, in the limiting position, the instantaneous rate at the point x; or, in symbols,
at some particular point.
For the particular function under consideration, therefore, = 2x.
2—5, Geometric Interpretation of the Derivative. Referring to the function discussed in §2—4, let P(x1,y1) and Q(x2,y2) be any two neighboring points on the curve. From the figure below, the slope of the secant PQ is given by the ratio . Now, as Δx → 0, point Q will move along the curve toward P as a limit; the secant PQ will approach the position of the tangent PT as its limiting position; and the angle RPQ will approach the angle RPT, or ϕ, as its limiting position.
Inasmuch as gives the slope of the secant PQ, the limit of this ratio, as Δx → 0, gives the slope of the tangent to the curve at the point P. In other words, the slope of the tangent to the curve at the point P is the value of
For this particular function, y = x2, we see that the slope of the tangent at any point is given by
or = 2x.
For example, at the point where x = 2, the slope of the tangent equals 2(2), or 4; at the point where x = 3, the slope equals 2(3), or 6; at x = 10, the slope equals 2(10), or 20; etc.
2—6. The Derivative as a Slope. The conclusions of the preceding paragraph may now be generalized. Consider any function y = f(x); P(x0,y0) any particular point on the curve; and Δx an arbitrary increment of the independent variable. Then the corresponding increment of the dependent variable, or of the function, is
Δy = f(x0 + Δx) − f(x0) = RQ.
The average rate of change within the interval Δx is given by
But = is also the slope of the secant PQ.
Now as Δx → 0, so that the interval Δx “closes down” about the point x, the limiting value of the ratio becomes the derivative . The derivative thus represents not only the instantaneous rate of change at the point where x = x0, but also represents the slope of the tangent to the curve at the point x = x0. This, of course, means that the derivative represents the slope of the curve at that point, since we learned that the slope of a curve at any point is given by the slope of the tangent to the curve at that point.
The sense in which the tangent line is the “limit” of the secant lines should be clearly understood. To begin with, the tangent is a fixed line; secondly, the secant is a variable line, depending upon the value given to Δx (that is, for every value of Δx, except 0, there is a corresponding position of the secant); and in the third place, the angle between the tangent and secant, angle TPQ, can be made to become and remain as small as we please by taking Δxsufficiently small. The tangent line itself does not, in general, belong to the series of secant lines; it is a unique line, which bears a special relation to the series of secant lines. The tangent is the only line through P that has the property here described.
In conclusion, then, we may write:
where f′(x) is merely an alternative symbolic form for . Still a third symbol often used for the derivative is Dxy.