`f(x) = x^2 - 4x + 5` Find the derivative of the function by the limit process.

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Chapter 2, Review - Problem 3 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find derivative using limit definition, such that:

`f'(x)= lim_(Delta x -> 0) (f(x + Delta x) - f(x))/(Delta x)`

`f'(x) = lim_(Delta x -> 0) ((x + Delta x)^2 - 4(x+Delta x) + 5 - x^2 + 4x - 5)/(Delta x)`

`f'(x) = lim_(Delta x -> 0) (x^2 + 2x*Delta x + Delta^2 x - 4x - 4 Delta x + 5 - x^2 + 4x - 5)/(Delta x)`

Reducing like terms yields:

`f'(x) = lim_(Delta x -> 0) (2x*Delta x + Delta^2 x - 4 Delta x)/(Delta x)`

Simplify by `Delta x` :

`f'(x) = lim_(Delta x -> 0) 5(2x + Delta x - 4)`

Replacing 0 for `Delta x` yields:

`f'(x) = 2x - 4`

Hence, evaluating the limit of function using limit definition, yields `f'(x) = 2x - 4.`

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