4fed8e8aecad042e72000009 copy.png

$\frac{df}{dt} = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}$

This equation is the fundamental definition of the derivative of a function $f(t)$ with respect to $t$.

Let's break down each part:

1. $\frac{df}{dt}$: This notation represents the derivative of the function $f$ with respect to the variable $t$. It signifies the instantaneous rate of change of $f$ as $t$ changes.

2. $\frac{f(t+h) - f(t)}{h}$: This is called the difference quotient. It represents the average rate of change of the function $f$ over the interval $[t, t+h]$ (or $[t+h, t]$ if $h$ is negative).

   • $f(t+h)$ is the value of the function at a slightly shifted point.
   • $f(t)$ is the value of the function at the original point.
   • $h$ is the small change in the independent variable $t$.

3. $\lim_{h \to 0}$: This is the limit as $h$ approaches zero. By taking this limit, we are making the interval over which we calculate the average rate of change infinitesimally small. As $h$ gets closer and closer to zero, the average rate of change over that shrinking interval approaches the instantaneous rate of change at the point $t$.

In essence: The derivative $\frac{df}{dt}$ gives us the instantaneous rate of change of the function $f$ at a specific point $t$. Geometrically, it represents the slope of the tangent line to the graph of $f(t)$ at the point $(t, f(t))$.

This definition is central to differential calculus and is used to define concepts like velocity (the derivative of position with respect to time), acceleration (the derivative of velocity), and many other rates of change in science and engineering.