Symbols instructing differentiation are : or D; Symbols indicating differentiation to the variable are: or ; x Dx Differentiation with respect to the variable are indicated by: or ; y Dy Differentiation of with respect to are indicated by: f(x) x or or Dx[f(x)] or (x) f(x) Differentiation of with respect to are indicated by: y x or or Dx[y] or y 8.2 Rules for differentiation The following are the commonly used rules for differentiation. 1. If = ; where is a real number; then : y xn n = nxn–1 2. [f(x) = g(x)] f(x) g(x) 3. = ; where is a constant. k 4. = 0 ; where is a constant. k Worked examples 1. Use the rules of differentiation to calculate the derivatives of the following functions with respect to x. a) = b) = ½ y x3 f(x) x Use rule 1: = 3 ; = 2 Use rule 1: = ½ ; = n n–1 n n –1 –1/2 = 3x3-1 = 3x2 = c) 3x2 6 d) 3 2x y = y = – – – 6x 3 2x 4x-2 = y = – – = 0 4(–2x–3) –2 – = 0 2 + 8x–3 – = 2 + – 2. Use the rules of differentiation to differentiate the following function with respect to i.e: = + + m; y mx3 m2x m3x – Solution: = + 2mx 3m2x Dm x3 – 3. Calculate (t), using the rules of differentiation, if f(t) = Solution: f(t) = = 3 (t) = 3(. ) (t) = Activity 6 1. Calculate, in the following using the rules of differentiation. a) = b) = 99 y x6 y c) 4x3 2x2 4x 3 d) = y = + + y – 2. Differentiate, the following with respect to using the rules of differentiation. x,
Symbols instructing differentiation are : or D; Symbols indicating differentiation to the variable x are: or Dx ; Differentiation with respect to the variable y are indicated by: or Dy ; Differentiation of f(x) with respect to x are indicated by: f(x) or or Dx[f(x)] or (x) Differentiation of y with respect to x are indicated by: y or or Dx[y] or 8.2 Rules for differentiation The following are the commonly used rules for differentiation. 1. If y = xn ; where n is a real number; then : = nxn–1 2. [f(x) g(x)] = f(x) g(x) 3. = ; where k is a constant. 4. = 0 ; where k is a constant. Worked examples 1. Use the rules of differentiation to calculate the derivatives of the following functions with respect to x. a) y = x3 Use rule 1: n = 3 ; n–1 = 2 = 3x3-1 = 3x2 b) f(x) = x ½ Use rule 1: n = ½ ; n –1 = –1/2 = c) y = 3x2 – 6 = 6x d) y = 3 – 2x – y = 3 – 2x – 4x-2 = 0 –2 – 4(–2x–3) = 0 – 2 + 8x–3 = – 2 + 2. Use the rules of differentiation to differentiate the following function with respect to m; i.e: y = mx3 + m2x – m3x + Solution: Dm = x3 + 2mx – 3m2x 3. Calculate (t), using the rules of differentiation, if f(t) = Solution: f(t) = = 3 (t) = 3(. ) (t) = Activity 6 1. Calculate, in the following using the rules of differentiation. a) y = x6 b) y = 99 c) y = 4x3 + 2x2 – 4x + 3 d) y = 2. Differentiate, the following with respect to x, using the rules of differentiation.