`f(x) = x^2 + x - 3` Find the derivative of the function by the limit process.

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By limit process, the derivative of a function f(x) is :-

f'(x) = lim h --> 0 [{f(x+h) - f(x)}/h]

Now, the given function is :-

f(x) = (x^2) + x - 3

thus, f'(x) = lim h --> 0  [{{(x+h)^2} + (x+h) - 3} - ((x^2) + x - 3)}/h]

or, f'x) = lim h --> 0 [{{(x+h)^2} - (x^2) + h}/h]

or, f'(x) = lim h --> 0 [{(x^2) + (h^2) + 2xh + h - (x^2)}/h]

or,  f'(x) = lim h --> 0 [{(h^2) + h + 2xh}/h] = [h + 1 + 2x]

putting the value of h = 0 in the above  expression we get

f'(x) = 2x + 1

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