Q.Program 6-4: Program to find the larger of the two pre-specified numbers.

Variable Difference — The Intuition

Imagine you're watching two runners on a track. One runs fast, the other slow. The gap between them keeps changing — sometimes it grows, sometimes it shrinks. That changing gap is a variable difference.

Now think about a simpler case: two numbers, say 10 and 3. Their difference is 7 — fixed, constant. But what if the numbers themselves are changing? Then the difference between them changes too. That's all "variable difference" means: the difference between two quantities that are themselves varying.

The Precise Statement

Let two quantities depend on some common parameter — often time $t$, or a variable $x$. Call them $f(x)$ and $g(x)$. Their difference is:

Since $f$ and $g$ change with $x$, $D$ changes too. That's a variable difference.

The key question is: how fast does this difference change? That's where calculus enters. The rate of change of the difference is simply:

In words: the rate at which the gap changes equals the difference of the individual rates.

Why This Matters

In Indian exam problems, you'll see this in two common forms:

1. Relative speed (motion problems)

Two trains leave stations at different times, moving at different speeds. The distance between them is a variable difference. Its rate of change is the difference of their speeds (if moving in the same direction) or the sum (if opposite).

2. Increasing/decreasing functions

If $f(x) > g(x)$ for all $x$, and you want to know whether the gap widens or narrows, you check the sign of $f'(x) - g'(x)$. Positive means the gap grows; negative means it shrinks.