Derivative of simple algebraic or polynomial functions.

The derivative and calculations on finding derivative of simple algebraic functions or polynomial functions is given below:

1> Derivative of Constant function or derivative of f(x)=y=c (c is a constant)

Let Δx be a small increment in x and Δy be corresponding increment in y. Then,

$f(x+\Delta x)=y+\Delta y=c$
or, $\Delta y=c-y$
$= c - c$ $=0$

and , $\dfrac{\Delta y}{\Delta x}=0$

Thus , $\dfrac{dy}{dx}=\displaystyle\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}=0$

2> Derivative of Identity function or derivative of f(x)=y=x

Let Δx be a small increment in x and Δy be corresponding increment in y. Then,
$f(x+\Delta x)=y+\Delta y=x+\Delta x$
or, $\Delta y=x+\Delta x-y$
$=\Delta x$

and , $\dfrac{\Delta y}{\Delta x}=1$

Thus, $\dfrac{dy}{dx}=\displaystyle \lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}=1$

3> Derivative of simple Quadratic function or derivative of f(x)=y=x²

Let Δx be a small increment in x and Δy be corresponding increment in y. Then,
$f(x+\Delta x)=y+\Delta y=(x+\Delta x)^2$
$=x^2+2.x.\Delta x+\Delta x^2$
or, $\Delta y=x^2+2.x.\Delta x+\Delta x^2-y$
$=\Delta x(2x+\Delta x)$

and, $\dfrac{\Delta x}{\Delta y}=2.x+\Delta x$

Thus, $\dfrac{dy}{dx}=\displaystyle \lim_{\Delta x\to 0}2x+\Delta x=2x$

4> Derivative of Simple cubic function or derivative of f(x)=y=x³

Let Δx be a small increment in x and Δy be corresponding increment in y. Then,
$f(x+\Delta x)= y+\Delta y=(x+\Delta x)^3$
$y+ \Delta y=x^3+3x^2.\Delta x+3x.\Delta x^2+\Delta x^3$

or, $\Delta y=x^3+3x^2.\Delta x+3x.\Delta x^2+\Delta x^3-y$
$=3x^2.\Delta x+3x. \Delta x^2+\Delta x^3$
$=\Delta x(3x^2+3x.\Delta x+\Delta x^2)$

and, $\dfrac{dy}{dx}= 3x^2+3x.\Delta x+\Delta x^2$

Thus, $\dfrac{dy}{dx} = \displaystyle\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}=\displaystyle\lim_{\Delta x\to 0} 3x^2+3x.\Delta x+\Delta x^2 = 3x^2$