Definition:Derivative

Let $$f: \mathbb{R} \to \mathbb{R}$$ be a continuous function.

If $$y = f \left({x}\right)$$, then the derivative of $$y$$ with respect to $$x$$ is defined as:

$$\mathbf {Define:} \ \frac{dy}{dx} \ \stackrel {\mathbf {def}} {=\!=} \ \lim_{\delta x \to 0} \frac {f \left({x + \delta x}\right) - f \left({x}\right)} {\delta x}$$

if that limit exists.

It can also be denoted by $$y'$$, $$\frac{df}{dx}$$, $$\frac{d}{dx} \left({f}\right)$$ or $$f^{\prime} \left({x}\right)$$.

Note that it can also be written, and frequently is, as $$\frac{dy}{dx} = \lim_{h \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} {h}$$.

This makes the notation simpler, but can tend to obscure the fact that the derivative involves the change of a value of a function for a "small change" in $$x$$.

Yet another particularly elegant way of expressing the same concept is $$\frac{dy}{dx} = \lim_{x \to w} \frac {f \left({w}\right) - f \left({x}\right)} {w - x}$$.

Second Derivative
Let $$f: \mathbb{R} \to \mathbb{R}$$ be a continuous function.

Let $$y = f \left({x}\right)$$.

Let $$y'$$ be the derivative of $$y$$ with respect to $$x$$.

Then the second derivative of $$y$$ with respect to $$x$$ is defined as $$\frac{dy'}{dx}$$, the derivative of the derivative (which, in this context, can be referred to as the "first derivative"):

$$\mathbf {Define:} \ \frac{dy'}{dx} \ \stackrel {\mathbf {def}} {=\!=} \ \lim_{\delta x \to 0} \frac {f' \left({x + \delta x}\right) - f' \left({x}\right)} {\delta x}$$

if that limit exists.

It can also denoted by $$y''$$, $$\frac{d^2y}{dx^2}$$, $$\frac{d^2f}{dx^2}$$, $$\frac{d^2}{dx^2} \left({f}\right)$$ or $$f^{\prime \prime} \left({x}\right)$$.

Higher Derivatives
Higher derivatives are defined in similar ways.

Differentiation
The process of obtaining the derivative of a function $$f$$ with respect to $$x$$ is known as differentiation (of $$f$$) with respect to $$x$$, or differentiation w.r.t. $$x$$