Geometrical Meaning of Differentiation

It is useful to consider what geometrical meaning can be given to the derivative.

In the first place, any function of , such, for example, as , or , or , can be plotted as a curve; and nowadays many students are familiar with the process of curve-plotting. Various tools are available for plotting curves, such as graphing calculators, Wolfram Alpha,¹ MATLAB, Python, or even Microsoft Excel.

Let , in the next figure, be a portion of a curve plotted with respect to the and axes. Consider any point  on this curve with coordinates (i.e. the abscissa of the point is  and its ordinate is ).

Now observe how  changes when  is varied. If is made to increase by a small increment , to the right, it will be observed that also (in this particular curve) increases by a small increment  (because this particular curve happens to be an ascending curve). Then the ratio of  to  is a measure of the degree to which the curve is sloping up between the two points  and . As a matter of fact, it can be seen on the figure that the curve between and  has many different slopes, so that we cannot very well speak of the slope of the curve between and . If, however, and  are so near each other that the small portion  of the curve is practically straight, then it is true to say that the ratio  is the slope of the curve along . The straight line  produced on either side touches the curve along the portion  only, and if this portion is indefinitely small, the straight line will touch the curve at practically one point only, and be therefore a tangent to the curve.

This tangent to the curve has evidently the same slope as , so that is the slope of the tangent to the curve at the point  for which the value of  is found.

We have seen that the short expression “the slope of a curve” has no precise meaning, because a curve has so many slopes—in fact, every small portion of a curve has a different slope. “The slope of a curve at a point” is, however, a perfectly defined thing; it is the slope of a very small portion of the curve situated just at that point; and we have seen that this is the same as “the slope of the tangent to the curve at that point.”

Observe that is a short step to the right, and the corresponding short step upwards. These steps must be considered as short as possible—in fact indefinitely short,—though in diagrams we have to represent them by bits that are not infinitesimally small, otherwise they could not be seen.

We shall hereafter make considerable use of this circumstance that represents the slope of the curve at any point.

If a curve is sloping up at  at a particular point, as in the next figure, and  will be equal, and the value of .

If the curve slopes up steeper than  (next figure),  will be greater than .

If the curve slopes up very gently, as in the next figure,  will be a fraction smaller than .

For a horizontal line, or a horizontal place in a curve, , and therefore .

If a curve slopes downward, as in the next figure,  will be a step down, and must therefore be reckoned of negative value; hence  will have negative sign also.

If the “curve” happens to be a straight line, like that in the following figure, the value of  will be the same at all points along it. In other words its slope is constant.

If a curve is one that turns more upwards as it goes along to the right, the values of  will become greater and greater with the increasing steepness, as in the following figure.

If a curve is one that gets flatter and flatter as it goes along, the values of  will become smaller and smaller as the flatter part is reached, as in the next figure.

If a curve first descends, and then goes up again, as in the next figure, presenting a concavity upwards, then clearly will first be negative, with diminishing values as the curve flattens, then will be zero at the point where the bottom of the trough of the curve is reached; and from this point onward will have positive values that go on increasing. In such a case  is said to pass by a minimum. The minimum value of  is not necessarily the smallest value of , it is that value of  corresponding to the bottom of the trough; for instance, in the next figure, the value of  corresponding to the bottom of the trough is , while  takes elsewhere values which are smaller than this. The characteristic of a minimum is that  must increase on either side of it.

Note—For the particular value of  that makes  a minimum, the value of .

If a curve first ascends and then descends, the values of  will be positive at first; then zero, as the summit is reached; then negative, as the curve slopes downwards, as in the next figure. In this case  is said to pass by a maximum, but the maximum value of  is not necessarily the greatest value of . In the above figure, the maximum of  is , but this is by no means the greatest value can have at some other point of the curve.

Note—For the particular value of  that makes  a maximum, the value of .

If a curve has the peculiar form of the following figure, the values of  will always be positive; but there will be one particular place where the slope is least steep, where the value of  will be a minimum; that is, less than it is at any other part of the curve.

If a curve has the form of the following figure, the value of  will be negative in the upper part, and positive in the lower part; while at the nose of the curve where it becomes actually perpendicular, the value of will be infinitely great.

In summary:

Now that we understand that  measures the steepness of a curve at any point, let us turn to some of the equations which we have already learned how to differentiate.

Example 10.1. As the simplest case take this:

It is plotted out in the following figure, using equal scales for  and . If we put , then the corresponding ordinate will be ; that is to say, the “curve” crosses the -axis at the height . From here it ascends at ; for whatever values we give to  to the right, we have an equal  to ascend. The line has a gradient of  in .

Now differentiate , by the rules we have already learned, and we get .

The slope of the line is such that for every little step  to the right, we go an equal little step  upward. And this slope is constant—always the same slope.

Example 10.2. Take another case: We know that this curve, like the preceding one, will start from a height  on the -axis. But before we draw the curve, let us find its slope by differentiating; which gives . The slope will be constant, at an angle, the tangent of which is here called . Let us assign to  some numerical value—say . Then we must give it such a slope that it ascends  in ; or will be  times as great as ; as magnified in the following figure.

So, draw the line in the next figure at this slope.

Now for a slightly harder case.

Example 10.3. Let

Again the curve will start on the -axis at a height  above the origin.

Now differentiate. [If you have forgotten, turn back; or, rather, don’t turn back, but think out the differentiation.]

This shows that the steepness will not be constant: it increases as  increases. At the starting point , where , the curve (next figure) has no steepness—that is, it is level. On the left of the origin, where has negative values,  will also have negative values, or will descend from left to right, as in the Figure.

Let us illustrate this by working out a particular instance. Taking the equation and differentiating it, we get Now assign a few successive values, say from  to , to ; and calculate the corresponding values of  by the first equation; and of  from the second equation. Tabulating results, we have:

Then plot them out in two curves, Fig. 10.18 and Fig. 10.19; in Fig. 10.18 plotting the values of  against those of  and in Fig. 10.19 those of against those of . For any assigned value of , the height of the ordinate in the second curve is proportional to the slope of the first curve.

If a curve comes to a sudden cusp, as in the following figure, the slope at that point suddenly changes from a slope upward to a slope downward. In that case will clearly undergo an abrupt change from a positive to a negative value.

The following examples show further applications of the principles just explained.

Example 10.4. (a) Find the slope of the tangent to the curve at the point where . (b) Find the angle which this tangent makes with the curve .

Solution. (a) The slope of the tangent is the slope of the curve at the point where they touch one another; that is, it is the of the curve for that point. Here and for , , which is the slope of the tangent and of the curve at that point. The tangent, being a straight line, has for equation , and its slope is , hence . Also if , ; and as the tangent passes by this point, the coordinates of the point must satisfy the equation of the tangent, namely so that and ; the equation of the tangent is therefore (see the following figure).

(b) Now, when two curves meet, the intersection being a point common to both curves, its coordinates must satisfy the equation of each one of the two curves; that is, it must be a solution of the system of simultaneous equations formed by coupling together the equations of the curves. Here the curves meet one another at points given by the solution of

that is,

This equation has for its solutions and (see the next figure). The slope of the curve at any point is

For the point where , this slope is zero; the curve is horizontal. For the point where hence the curve at that point slopes downwards to the right at such an angle  with the horizontal that ; that is, at  to the horizontal.

The slope of the straight line is ; that is, it slopes downwards to the right and makes with the horizontal an angle  such that ; that is, an angle of . It follows that at the first point the curve cuts the straight line at an angle of , while at the second it cuts it at an angle of (see the next figure).

Example 10.5. A straight line is to be drawn, through a point whose coordinates are , , as tangent to the curve . Find the coordinates of the point of contact.  
Note.—-the point does not lie on the curve .

Solution. The slope of the tangent must be the same as the of the curve; that is, .

The equation of the straight line is , and as it is satisfied for the values , , then ; also, its [since is the tangent line, its slope, , must be the same as ].

The and the of the point of contact must also satisfy both the equation of the tangent and the equation of the curve.

We have then four equations in , , , .

Equations (i) and (ii) give .

Replacing and  by their value in this, we get which simplifies to , the solutions of which are: and . Replacing in (i), we get and respectively; the two points of contact are then , , and , (see the following figure).

Note.—In all exercises dealing with curves, students will find it extremely instructive to verify the deductions obtained by actually plotting the curves.

Exercises