Derivative of a Function
Let $$y = f(x)$$ be a given function of $$x$$. Give to $$x$$ a small increment $$\Delta x$$ and let the corresponding increment of $$y$$ by $$\Delta y$$, so that when $$x$$ becomes $$x + \Delta x$$, then $$y$$ becomes $$y + \Delta y$$ and we have:
$$y + \Delta y = f(x + \Delta x)$$
$$\therefore \Delta y = f(x + \Delta x) – f(x)$$
Dividing both sides by $$\Delta x$$, then
$$\frac{{\Delta y}}{{\Delta x}} = \frac{{f(x + \Delta x) – f(x)}}{{\Delta x}}$$
Taking limit of both sides as $$\Delta x \to 0$$
$$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}}\mathop { = \lim }\limits_{\Delta x \to 0} \frac{{f(x + \Delta x) – f(x)}}{{\Delta x}}$$
Thus, if $$y$$ is the function of $$x$$, then $$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{f(x + \Delta x) – f(x)}}{{\Delta x}}$$ is called the derivative or the differential coefficient of the function or the derivative of $$f(x)$$ with respect to $$x$$ and is denoted by $$f'(x)$$, $$y’$$, $$Dy$$ or $$\frac{{dy}}{{dx}}$$.
Note: It may be noted that the derivative of the function $$f(x)$$ with respect to the variable $$x$$ is the function $$f’$$ whose value at $$x$$ is $$f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) – f(x)}}{h}$$ provided the limit exists, is called the derivative where $$h$$ is the increment.
Below we list the notations for derivative of $$y = f(x)$$ used by different mathematicians in a table.
Name of Mathematician

Leibniz

Newton

Lagrange

Cauchy

Notation for Derivative

$$\frac{{dy}}{{dx}}$$or$$\frac{{df}}{{dx}}$$  $$y’$$  $$f'(x)$$  $$Df(x)$$ 
Remarks:
I. The student should observe the difference between $$\frac{{\Delta y}}{{\Delta x}}$$ and $$\frac{{dy}}{{dx}}$$. Where $$\frac{{\Delta y}}{{\Delta x}}$$ is the quotient of the increment of $$y$$ and $$x$$ that is its numerator and denominator can be separated, but $$\frac{{dy}}{{dx}}$$ is a single symbol for the limiting value of the fraction $$\frac{{\Delta y}}{{\Delta x}}$$ when $$\Delta y$$ and $$\Delta x$$ can be separated.
II. $$\frac{d}{{dx}}$$ attached to any function meaning its differential coefficient with respect to $$x$$.
III. The phrase “with respect to” will often be abbreviated into w.r.t
Differentiation:
The process of finding the differential coefficient of a function or a process for finding the rate at which one variable quantity changes with respect to another is called differentiation.
Four Steps in Differentiation:
In the given function
$$y = f(x)$$
1. Change $$x$$ to $$x + \Delta x$$ and $$y$$ to $$y + \Delta y$$
i.e. $$y + \Delta y = f(x + \Delta x)$$
2. Find $$\Delta y$$ by subtraction
i.e. $$\Delta y = f(x + \Delta x) – y$$
$$\Delta y = f(x + \Delta x) – f(x)$$
3. Divide both sides by $$\Delta x$$
i.e. $$\frac{{\Delta y}}{{\Delta x}} = \frac{{f(x + \Delta x) – f(x)}}{{\Delta x}}$$
4. Find the limit of $$\frac{{\Delta y}}{{\Delta x}}$$ where $$\Delta x \to 0$$
i.e. $$\mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}}\mathop { = \lim }\limits_{\Delta x \to 0} \frac{{f(x + \Delta x) – f(x)}}{{\Delta x}}$$
Employing the above mentioned four steps in determining the derivative means finding the differential coefficient “by definition” or “by first principle” or “by abinitio method”.