In mathematics, the derivative shows the sensitivity of change of a function’s output with respect to the input. But what exactly does that mean?
From Physics class in high school, we all learnt to calculate the average velocity over an interval, which is the change of distance (or displacement) over change in time, i.e.
$$ v = \frac{\Delta x}{\Delta t} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} $$
where $s$ is a function that returns the distance $x$ at a certain time $t$.
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This allows us to find the slope (rate of change) between two points, but what if I want to find the slope at a single point? To do so, you will have to make the difference in the time $t$ so small and measure the corresponding change in distance $x$. This gives us the derivative, or often described as the “instantaneous rate of change”
$$ s'(t) = \frac{dx}{dt} = \lim_{\Delta t\to 0}\frac{s(t + \Delta t) - s(t)}{\Delta t} $$
In other words, the derivative of $s$ equals to the limit of the average rate of change in distance over time $\frac{s(t + \Delta t) - s(t)}{\Delta t}$ as the difference in time $\Delta t$ approaches zero.
Geometrically, the derivative at a point represents the slope of the tangent line to the curve representing the function at that specific point.
Note that it is kinda paradoxical to call it the instantaneous rate of change because how can change be instantaneous if it is something measured over an interval? Perhaps a better way to understand it would that it is the best possible linear approximation of the change at a certain point.
Given a function $f$ where
$$ y = f(x) $$
how do we denote its derivative? Both the notations below are accepted and used depending on the given context