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

The Conclusion:

If we analysis above four examples and also analysis the derivative of higher degree of functions

Then we can see the following result:

Derivative of simple algebraic functions or polynomial functions

like function f(x)=y=xn

is n.xn-1

or,\dfrac{dx^n}{dx}=n.x^{n-1}



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