`f(x) = x^3 - 12x` Find the derivative of the function by the limit process.

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Chapter 2, 2.1 - Problem 19 - Calculus of a Single Variable (10th Edition, Ron Larson).
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hkj1385 | (Level 1) Assistant Educator

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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^3) - 12x 

thus, f'(x) = lim h --> 0  [{{(x+h)^3} - 12(x+h)} - ((x^3) - 12x)}/h]

or, f'x) = lim h --> 0 [{{(x+h)^3} - (x^3) - 12h}/h]

or, f'(x) = lim h --> 0 [{(x^3) + (h^3) + 3x(h^2) + 3(x^2)h - 12h - (x^3)}/h]

or,  f'(x) = lim h --> 0 [{(h^3) -12h + 3x(h^2) + 3(x^2)h}/h]

  = [(h^2) + 3xh + 3(x^2) - 12]

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

f'(x) = 3(x^2) - 12

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