Question

How do you find the derivative of `f(x)= x^2 -5x + 3` using the limit definition?

Answer 1

`lim_(Deltax->0) [f(x+Deltax) - f(x)]/(Deltax)`

`f'(x)=lim_(Deltax->0) [2x+Deltax-5]=2x-5`

Explanation:

Given: `f(x) = x^2-5x+3`

Required: Derivative using limits

Solution Strategy:
Definition: `(df(x))/(dx)=lim_(Deltax->0) [f(x+Deltax) - f(x)]/(Deltax)`
A) Evaluate `f(x)|_(x=x-Deltax)`
B) Subtract `f(x)` and simplify

A) `f(x)|_(x=x+Deltax)= (x+Deltax)^2-5(x+Deltax)+3`
`=cancelx^2+2xDeltax+Deltax^2-cancel(5x)- 5Deltax+cancel3`
`- cancelx^2+cancel(5x)-cancel3`

`f'(x)=lim_(Deltax->0) [2xcancel(Deltax)+cancel(Deltax^2)^(Deltax)-5cancel(Deltax)]/cancel(Deltax)`

`f'(x)=lim_(Deltax->0) [2x+Deltax-5]=2x-5`

Good luck!