Definition of a Derivative

The slope of a straight line is an important concept in traditional algebra, and combining it with the idea of a limit results in one of the most fundamental notions in calculus: the instantaneous slope of a curve, or the derivative. The process of finding a derivative is differentiation, not to be confused with "deriving" a formula.

Visualize any smooth curve given by `y=f(x)`. Clearly, picking any two points on the curve `\left(x_0, f\left(x_0\right)\right)` and `\left(x_1, f\left(x_1\right)\right)` with `x_0 < x_1` will allow a line to be drawn between them, where the slope is `\dfrac{f\left(x_1\right)-f\left(x_0\right)}{x_1-x_0}`. Visually, it is clear that slowly moving `x_1` towards `x_0` will begin to approximate the instantaneous slope of the curve at `x=x_0`, and can be formalized as such:

`\displaystyle\boxed{\frac{\mathrm df}{\mathrm dx}=f'(x)=\dot f(x)=D_xf(x)\overset{\mathrm{def}}{=}\lim_{x\to x_0}\frac{f(x)-f\left(x_0\right)}{x-x_0}=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}}`

where

• `\dfrac{\mathrm df}{\mathrm dx}` is known as Leibniz notation (often useful for conceptual understanding)


• `f'(x)` (pronounced "f prime") is prime notation (useful for brevity)


• `\dot f(x)` is dot notation (used primarily in physics, with respect to time derivatives)


• `D_xf(x)` is operator notation (used most commonly when studying differential equations)

Each of these notations is used in different contexts to represent the derivative of `f`, the instantaneous slope of the tangent line to the curve `y=f(x)`.