# 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 ab-initio method”.