Differentiate `f(x)=x^2-5x+3` from first principle.    

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To differentiate the function from first principles, we need to evaluate the limit:


Consider the numerator of the limit.


`=(x+h)^2-5(x+h)+3-(x^2-5x+3)`   expand brackets

`=x^2+2xh+h^2-5x-5h+3-x^2+5x-3`   collect like terms

`=2xh-5h+h^2`   factor the h


Now put into the numerator of the limit to get:

`f'(x)=lim_{h->0}{h(2x-5+h)}/h`   cancel common factor

`=lim_{h->0}(2x-5+h)`     now take the limit


The derivative of the function is `f'(x)=2x-5` .

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