Question

Find the derivative of function y=4x^2-9x-6 using the limiting process . Show your work including: difference quotient , steps to simplify, how you set up the limit , and the derivative

Answer 1

By the definition of the derivative,

`f'(x)=\displaystyle\lim_(h\to0)\frac{f(x+h)-f(x)}h`

`\implies y'=\displaystyle\lim_(h\to0)\frac{(4(x+h)^2-9(x+h)-6)-(4x^2-9x-6)}h`

Simplify the numerator:

`4(x+h)^2-9(x+h)-6=4(x^2+2xh+h^2)-9(x+h)-6`

`=4x^2+8xh+4h^2-9x-9h-6`

Subtracting
`4x^2-9x-6` removes all the terms here not involving
`h`, so the limit reduces to

`y'=\displaystyle\lim_(h\to0)\frac{8xh+4h^2-9h}h`

Cancel the factor
`h` in the numerator and denominator:

`y'=\displaystyle\lim_(h\to0)(8x+4h-9)`

As
`h\to0`, we're left with

`\boxed{y'=8x-9}`