## 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 Δ*y* and Δ*x* 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*(*x*_{1},*y*_{1}) on the curve *y* = *x*^{2}, and *Q* (*x*_{2},*y*_{2}), another point on the curve, nearby. By making Δ*x*, or *x*_{2} − *x*_{1}, small enough, we can make the increment Δ*y*, or *y*_{2} − *y*_{1}, as small as we please. In short, as Δ*x* → 0, Δ*y* also approaches zero.

Now, it is easy to show that the average rate of change of *y* during the interval from *x*_{1} to *x*_{2} is given by

= 2*x* + Δ*x*.

For,*y* = *x*^{2}.(1)

*y* + Δ*y* = (*x* + Δ*x*)^{2},

or*y* + Δ*y* = *x*^{2} + 2*x*·Δ*x* + (Δ*x*)^{2}.(2)

Subtracting (1) from (2):

Δ*y* = 2*x*·Δ*x*+ (Δ*x*)^{2}.(3)

Dividing (3) by Δ*x*:

= 2*x* + Δ*x*.(4)

We may now let Δ*x* approach zero as a limit; the average rate for the interval Δ*x* 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, = 2*x*.

**2—5, Geometric Interpretation of the Derivative.** Referring to the function discussed in §2—4, let *P*(*x*_{1},*y*_{1}) and *Q*(*x*_{2},*y*_{2}) 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 Δ*x* → 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 Δ*x* → 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* = *x*^{2}, we see that the slope of the tangent at any point is given by

or = 2*x*.

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*(*x*_{0},*y*_{0}) any particular point on the curve; and Δ*x* an arbitrary increment of the independent variable. Then the corresponding increment of the dependent variable, or of the function, is

Δ*y* = *f*(*x*_{0} + Δ*x*) − *f*(*x*_{0}) = *RQ*.

The average rate of change within the interval Δ*x* is given by

But = is also the slope of the secant *PQ*.

Now as Δ*x* → 0, so that the interval Δ*x* “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* = *x*_{0}, but also represents the slope of the tangent to the curve at the point *x* = *x*_{0}. 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 Δ*x* (that is, for every value of Δ*x*, 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 Δ*x*sufficiently 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 *D _{x}y*.