Interpreting derivative vs change via ratio subtraction

Question

I have simple data in time like below:

time	A	B	rate (A/B)
June	50	100	0.5
July	65	300	0.2167

I want to find the change in the rate. From the above table it is clear that the rate decreased in the second month. Therefore shouldn't the gradient calculation be negative? Using gradient/derivative it turns out to be positive:

$$ =\frac{\delta A}{\delta B} =\frac{65-50}{300-100} = 0.075 $$

What is the interpretation of this gradient? And is this gradient a correct way to calculate the change in the rate column?

Answer 1

In the graph of $A$ plotted against $B,$ the gradient measures the rate of change of $A$ as $B$ increases.

• So, the more positive the gradient, the more $A$ increases per unit increase in $B.$
• While if the gradient is negative, then the smaller the gradient's absolute value, the less $A$ decreases per unit increase in $B.$

You can think of this gradient is a measure of the correlation between quantities $A$ and $B.$

On the other hand, if you require the (separate) rates of change of $A$ and $B$ (per unit time), then you ought to be plotting instead the graph of ‘quantity’ with respect to time (in your example, the month). It will have two separate curves/lines corresponding to $A$ and $B$ respectively.

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Addendum in response to OP's follow-up query:

Note that the gradient is not $A\div B,$ but, rather, $(\text{change in }A) ÷ (\text{ change in }B).$

So for example, on a distance-time graph, at the point $(4\mathrm s, 80\mathrm m),$ the velocity/gradient is not necessarily $\frac{80}4=20\mathrm {m/s}.$

(A single point per se does not contain information about the slope at that point.)

Answer 2

The rate of change in the rate is simply (0.2167 - 0.5)/1 , where 1 signifies one month. It is indeed negative.