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 $$f^{\prime} \left({x}\right)$$.

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.