Question

Find derivative of the following, using first principle:- (2x+3)²​

Answer 1

Given: f(x) + (2x + 3)²​

To find: The derivative of the given function w.r.t. x using first principles.

Solution:

By the definition of first principles, the derivative of a function is given by,

$\rm f'(x) = \lim\limits_{h\to 0} \dfrac{f(x+h) - f(x)}{h}$

So using this we have,

$\rm f'(x) = \lim\limits_{h\to 0} \dfrac{(2(x+h) + 3)^2 - (2x+ 3)^2}{h}$

$\rm f'(x) = \lim\limits_{h\to 0} \dfrac{(2x+2h + 3)^2 - (2x+ 3)^2}{h}$

${\rm f'(x) = \lim\limits_{h\to 0} \dfrac{(2x)^2 +(2h)^2 + (3)^2 + 2(2x\cdot2h + 2h\cdot3 + 2x \cdot3) - (4x^2 + 12x + 9)}{h}}$

$\rm f'(x) = \lim\limits_{h\to 0} \dfrac{4x^2 +4h^2 +9+ 8xh + 12h + 12x -4x^2 -12x -9}{h}$

$\rm f'(x) = \lim\limits_{h\to 0} \dfrac{4h^2 + 8xh + 12h}{h}$

$\rm f'(x) = \lim\limits_{h\to 0} \dfrac{h(4h + 8x+ 12)}{h}$

$\rm f'(x) = \lim\limits_{h\to 0} (4h + 8x+ 12)$

$\rm f'(x) = 4(0) + 8x+ 12$

$\rm f'(x) = 8x+ 12$

Hence the derivative of the given function is 8x + 12.