Why is the operation of differentiation called "instantaneous change"?

Question

It is said that

Differentiation is an operation which finds the instantaneous change of the direction of a curve.

Why is the word change emphasized by the word instantaneous?

Answer 1

If by “change in direction” you mean “slope”, that’s precisely what a derivative is. It’s not emphasizing anything, that’s simply the definition of a derivative, as opposed to the average slope between two points.

For the average slope across two points $(x, f(x))$ and $((x+h), f(x+h))$, or the slope of the secant, you have

$$m = \frac{\Delta y}{\Delta x} = \frac{f(x+h)-f(x)}{h}$$

However, as you let $h \to 0$, the secant approaches a tangent line and you find the derivative at the point (hence the term “instantaneous”), so you get

$$m = \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$$

For instance, differentiating $f(x) = x^2$ gives $f’(x) = 2x$, meaning the slope of the tangent that touches the curve at $x$ will have the slope $2x$.

As a real-life example, if the displacement of a car is given by the same function $s(t) = t^2$, then the instantaneous velocity of the car at any $t$ will be $2t$. As an example, at $t = 5$, $v = 10$.