Question

2) Differentiate 𝑓(𝑥) = 2𝑥2 − 5𝑥 + 2 using the first principle.

Answer 1

f(x) = 2 x^2 - 5 x + 2
f'(x) = 4x - 5
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I do not know what the first principle is but maybe
f(x+h) = 2 (x^2 + 2 x h + h^2) - 5 (x+h) + 2
= 2 x^2 + 4 x h + 2h^2 - 5 x - 5 h + 2

f(x+n) - f(x) = ( 2 x^2 + 4 x h + 2h^2 - 5 x - 5 h + 2) - (2 x^2 - 5 x + 2)
= 4 x h + 2 h ^2 -5 h
divide by h
= 4 x + 2 h - 5
limit as h -->0
= 4 x - 5
as we knew

Answer 2

Using First Principles:
dy/dx = Lim ( f(x+h) - f(x) )/h , as h ----> 0

for the rest, fill in the steps supplied by anonymous above.