a, then Article 6. It follows from Theorem 4 that if O(x) JIM 02(x) = x-+a r L We use this relation in the following way. lim O(x)I x->a Consider the identity f(x) g(x) = I[f(x) + g(x)]2 - 1[f(x) - g(x)]2. If we take the limit of both sides as x -- a, the proof is completed by use of the above relation and the fact that the limit of a sum is equal to the sum of the limits. lim fg = Jim 1[f + g] 2 - lim 4 [f - g] 2 4 = j[lim (f + g)]2 - 1[lim (f - g)]2 = 4[A +B]2- 4[A -B]2=AB.

Hence the derivative of the sum of a finite number of functions is equal to the sent of their derivatives. Illustration 4. Ify=2x4-x3-2x+7,then dx dx (2x4) - TX (x3) = 8x3 - 3x2 - 2 dx (2x) + d- (7) by III by I and II. Exercise Example 1. If f (x) = 3x3 -4 X2 39 find f' (x). \Vriting f (x) in the form f (x) = 3x - 4x-2, by III and II we Solution: have (3x) - dx (4x-2) = 3 + 8x-3. f' (x) 17X Example 2. For what values of x is the derivative of the function x3/2 X1/2 equal to zero? If y = x3/2 - x112, then Solutions: dy dx _ 3 x1/2 - 1 x-1/2.