2. Differentiate, = 3x + 6, from first principles. y Solution: Let = 3x + 6 f(x) Then; = 3(x + 6 f(x + h) + h) = 3x + 3h + 6 = = = = (x) = 3 3. Use first principles, to determine the derivative of the function defined by = g(x) –5. Solution: = is a horizontal line with gradient 0; g(x) –5 i.e. = g(x) –5x0 Then: = = 5 g(x + h) +h)0 –5(x – = = = = (x) = 0 Activity 5 Differentiate the following functions from principles. a) = b) 3x f(x) x f(x) = c) 2 d) 4 f(x) = x + f(x) = e) 3 6x f) 2px f(x) = f(x) = a + – g)** 3x2 h)** 4 f(x) = f(x) = x2 + Differentiation rules To calculate the derivative of, say = from first principles this may be a daunting f(x) xgoogle, task to the ordinary human being. To assist in accomplishing this and other similar tasks, rules for differentiation have been developed and accepted. Suppose that = , where is a real number y xn n = nxn–1 Then: 8.1 Notation The following are the commonly used symbols depicting differentiation of a function.
2. Differentiate, y = 3x + 6, from first principles. Solution: Let f(x) = 3x + 6 Then; f(x + h) = 3(x + h) + 6 = 3x + 3h + 6 = = = = (x) = 3 3. Use first principles, to determine the derivative of the function defined by g(x) = –5. Solution: g(x) = –5 is a horizontal line with gradient 0; i.e. g(x) = –5x0 Then: g(x + h) = –5(x +h)0 = – 5 = = = = (x) = 0 Activity 5 Differentiate the following functions from principles. a) f(x) = x b) f(x) = 3x c) f(x) = x + 2 d) f(x) = 4 e) f(x) = 3 – 6x f) f(x) = a + 2px g)** f(x) = 3x2 h)** f(x) = x2 + 4 Differentiation rules To calculate the derivative of, say f(x) = xgoogle, from first principles this may be a daunting task to the ordinary human being. To assist in accomplishing this and other similar tasks, rules for differentiation have been developed and accepted. Suppose that y = xn , where n is a real number Then: = nxn–1 8.1 Notation The following are the commonly used symbols depicting differentiation of a function.
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