Using the limit definition, how do you differentiate #f(x) =4+x-2x^2#?

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Answer 1 · 2016-05-21T14:26:20

#f'(x) =1-4x#,
see the derivation below.

By definition, the derivative of the function #f(x)# is
#f'(x) = lim_(Delta x -> 0) (f(x+Delta x)-f(x))/(Delta x)#

In our case it looks like

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

#= lim_(Delta x -> 0)(Delta x-4x Delta x-2(Delta x)^2)/(Delta x)=#

#= lim_(Delta x -> 0)(1-4x-2Delta x) =#

# = 1-4x - lim_(Delta x -> 0)(2Delta x) = 1-4x#