Calculating rate of change of a car [1]

**Calculating rate of change is the first of calculus’s two principal operations **

…the two principal symbols that are used in calculating… are:

*d*which merely means “a little bit of.”

Thus*dx*means a little bit of*x*; or*du*means a little bit of*u*.- ∫ which is merely a long S, and may be called (if you like) “the sum of.”

Thus ∫*dx*means the sum of all the little bits of*x*… That’s all.

In ordinary algebra which you learned at school, you were always hunting after some unknown quantity which you called *x* or *y*; or sometimes there were two unknown quantities to be hunted for simultaneously. You have now to learn to go hunting in a new way; the fox being now neither *x* nor *y*. Instead of this you have to hunt for this curious cub called *dy/dx*. The process of finding the value of *dy/dx* is called “differentiating.” But, remember, what is wanted is the value of this ratio when both *dy* and *dx* are themselves indefinitely small.

**Calculating rate of change means calculating a curve’s local slope **

It is useful to consider what geometrical meaning can be given to the differential coeffcient. In the first place, any function of x, such, for example, as *x ^{2}*, or

*√x*, or

*ax + b*, can be plotted as a curve…

Consider any point *Q* on this curve, where the abscissa of the point is *x* and its ordinate is *y*. Now observe how *y* changes when *x* is varied. If *x* is made to increase by a small increment *dx*, to the right, it will be observed that *y* also (in this particular curve) increases by a small increment *dy* (because this particular curve happens to be an ascending curve). Then the ratio of *dy* to *dx* is a measure of the degree to which the curve is sloping up between the two points *Q* and *T*. If… *Q* and *T* are so near each other that the small portion *QT* of the curve is practically straight, then it is true to say that the ratio *dy/dx* is the slope of the curve along *QT*.

**Calculating rate of change just takes a little algebra and common sense**

Now let us see how, on first principles, we can differentiate some simple algebraical expression. Let us begin with the simple expression *y = x ^{2}.* Let

*x*, then, grow a little bit bigger and become

*x + dx*; similarly,

*y*will grow a bit bigger and will become

*y + dy*. Then, clearly, it will still be true that the enlarged

*y*will be equal to the square of the enlarged

*x*. Writing this down, we have:

*y + dy = (x + dx) ^{2}*.

Doing the squaring we get:

*y + dy = x2 + 2x∙dx + (dx) ^{2}*.

Remember that *dx* meant a bit—a little bit—of *x*. Then *(dx) ^{2}* will mean a little bit of a little bit of

*x*; that is… it is a small quantity of the second order of smallness. It may therefore be discarded as quite inconsiderable in comparison with the other terms. Leaving it out, we then have:

*y + dy = x2 + 2x∙dx*.

Now *y = x ^{2}*; so let us subtract this from the equation and we have left

*dy = 2x∙dx*.

Dividing across by *dx*, we find

*dy/dx= 2x*.

Now this is what we set out to find. The ratio of the growing of *y* to the growing of *x* is, in the case before us, found to be *2x*.

- Keisler, H. Jerome.
*Elementary calculus: An infinitesimal approach*. Courier Corporation, 2012, p. xi. - Thompson, Silvanus Phillips.
*Calculus made easy*. 2nd ed., enlarged, MacMillan, 1914.