The Differential Coefficient Considered with Respect to its Magnitude

Excerpts

• The Differential Coefficient Considered with Respect to its Magnitude

  value of the variable, is, if we may use the phrase, the *index* of the change which the function would receive if the value of the variable were increased.

• The Differential Coefficient Considered with Respect to its Magnitude

  We do not take the increments themselves, but the proportion they bear to the changes in the variable which gave rise to them; so in estimating the rate of motion of two points, we either consider lengths described in the same time, or if that cannot be done, we judge, not by the lengths described in different times, but by the proportion of those lengths to the times, or the proportions of the units which express them.

• The Differential Coefficient Considered with Respect to its Magnitude

  In passing from $1000$ to $1003$, we have the logarithms $3$ and $3.0013009$, the above-mentioned ratio being $.0004336$, little more than a tenth of the former.

• The Differential Coefficient Considered with Respect to its Magnitude

  In the same way, if a point is moving, so that at the end of $1$ second it is at $3$ feet from a fixed point, and at the end of $2$ seconds it is at $5$ feet from the fixed point, we cannot say which way it is moving at the end of one second.

• The Differential Coefficient Considered with Respect to its Magnitude

  But if on adding any interval, *however small*, to the first second, the moving point does, during that interval, increase its distance from the fixed point, we can then certainly say that at the end of the first second the point is moving from the fixed point.

• The Differential Coefficient Considered with Respect to its Magnitude

  The objection becomes of less force as the increment diminishes, but always exists unless we take the limit of the ratio of the increments, instead of that ratio.

• The Differential Coefficient Considered with Respect to its Magnitude

  Every value of the variable, gives not only a different value to the function, but a different quantity of increase or decrease in passing to what we may call *contiguous* values, obtained by a given increase of the variable.

• The Differential Coefficient Considered with Respect to its Magnitude

  How well this answers to our previously formed ideas on such subjects as direction, velocity, and force, has already appeared.

Equations

Problems

No exercises in this chapter.