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

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