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Definition of Derivative The derivative of a function f given by y = f(x) with respect to x at any x in its domain is the number f(x) and is defined as f' (x) = lim/ ( x+h) -f(x) h-o h provided the limits exist. The derivative of a function is customarily denoted by f'(x), y' and dy. Example 1: Find the derivative of the function y = x2 using the above definition. Solution: y = x2 f'(x) = lim f(x + h) -f(x) h f(x + h)2 - x2 f' (x) = lim h-0 h x2 + 2xh+ h2 -x2 f'(x) = lim h-0 h 2xh + h2 f'(x) = lim h-0 h h (2x + h) f'(x) = lim h f' (x) = lim 2x + h fi(x) = 2x+0 f' (x) = 2x Example 2. Find the derivative of the following function using the definition of the derivative. f(x) = 2x2 - 16x + 35 Solution: f' (x) = lim f(x + h) -f(x) h 2(x + h)2 - 16(x + h)+35 - (2x2 - 16x + 35) f'(x) = lim h-0 2x2 + 4xh + 2h2 - 16x - 16h + 35 - 2x2 + 16x - 35) f'(x) = lim h 4xh + 2h2 - 16h f'(x) = lim h h(4x + 2h - 16) f'(x) = lim h f'(x) = lim 4x + 2h - 16 f'(x) = 4x +2(0)- 16 f'(x) = 4x - 16A. Evaluate the following functions using the definition of derivative. 1. y = 2x2 - 6x 2. y = 3x3 + 4x 3. y = v8x + 4 4. y= 2x r+1 5. y =