Differentiation Practice

1 Secondary 4 Additional Mathematics: Differentiation 1. Rules of Differentiation Rule Example (a) ! !" [𝑥#] = 𝑛𝑥#$%, where n is a real number. ! !" [𝑥&] = 4𝑥&$% = 4𝑥' (b) In particular, ! !" (𝑐) = 0, where c is a constant. ! !" (3) = 0 (c) ! !" [𝑘𝑥#] = 𝑘 ! !" [𝑥#] = 𝑘𝑛𝑥#$% , where k is a real number. ! !" [3𝑥(] = 5 ! !" [𝑥(] = 5(5𝑥($%) = 25𝑥& Examples (a) ! !" (𝑥' + 3𝑥) − 2𝑥 + 7) = ! !" (𝑥') + 3 ! !" (𝑥)) − 2 ! !" (𝑥) + ! !" (7) = 3𝑥'$% + 3(2𝑥)$%) − 2(1𝑥%$%) + 0 = 3𝑥) + 6𝑥 − 2 (b) ! !" 52𝑥' − ' "!6 = 2 ! !" (𝑥') − 3 ! !" (𝑥$)) = 2(3𝑥'$%) − 3(−2𝑥$') = 6𝑥) + 6𝑥$' = 6𝑥) + * "" (c) ! !" [(𝑥 − 2)(𝑥 + 1)] = ! !" (𝑥) − 𝑥 − 2) à *Expand before differentiation = 2𝑥)$% − 1𝑥%$% − 0 = 𝑥 − 1

2 2. Chain Rule: If y is a function of u and u is a function of x, that is y = f(u), where u =g(x), then d𝑦 d𝑥 = d𝑦 d𝑢 × d𝑢 d𝑥 3. Derivatives of some functions: ! !" (𝑥#) = 𝑛𝑥#$% ! !" (𝑘𝑒" ) = 𝑘𝑒" ! !" (𝑘 ln 𝑥) = + " ! !" (𝑘 sin 𝑥) = 𝑘 cos 𝑥 ! !" (𝑘 cos 𝑥) = −𝑘 sin 𝑥 ! !" (𝑘 tan 𝑥) = 𝑘sec) 𝑥 Examples of chain rule and derivatives of some functions: (a) ! !" (3𝑥 + 2) # ! = % ) (3𝑥 + 2)$# ! ! !" (3𝑥 + 2) à = % ) (3𝑥 + 2)$# !(3) = ' ) (3𝑥 + 2)$# ! = ' )√'"-) (b) ! !" (ln 𝑥)' = 3(ln 𝑥)) ! !" (ln 𝑥) à = 3(ln 𝑥)) 5 % "6 = '(/0 ")! " (c) ! !" (sin(𝑥) + 2𝑥)) = cos(𝑥) + 2𝑥) ! !" (𝑥) + 2𝑥) = (2𝑥 + 2) cos(𝑥) + 2𝑥) = 2(𝑥 + 1) cos(𝑥) + 2𝑥) 4. Product Rule: ! !" (𝑢𝑣) = 𝑢 !2 !" + 𝑣 !3 !" 5. Quotient Rule: ! !" 5 3 46 = 4$% $& $ 3$' $& 4! Differentiate from the outermost function to the innermost function, that is, • differentiate the square root function, then • differentiate 3x + 2 Differentiate from the outermost function to the innermost function, that is, • differentiate the cube function, then • differentiate lnx

3 Examples of Product rule and Chain rule: (a) ! !" F𝑒"! cos 𝑥G = 𝑒"! (− sin 𝑥) + 2𝑥𝑒"! (cos 𝑥) = 𝑒"! (2𝑥 cos 𝑥 − sin 𝑥) à Always factorize common multiple! (b) ! !" 5 "$% √%$)"6 = √%$)"(%)$("$%)6# !7(%$)")(# !($)) %$)" à Quotient rule = √%$)"-("$%)(%$)")(# ! %$)" = (%$)")(# ![(%$)")#-("$%)] %$)" à Factorize common factor (1 − 2𝑥)$# ! = %$)"-"$% (%$)") " ! à Bring down (1 − 2𝑥)$# ! = $" (%$)") " ! à Simplify 6. Problems on applications of differentiation include the following: (a) Gradients, tangents and normal. - !: !" refers to the gradient of the tangent at any point of the curve. - The gradient of the normal is = − % ;)*+,-+) as tangent ⊥ normal (b) Stationary points, increasing and decreasing functions. - At stationary points or inflection points, !: !" = 0. - For increasing functions, , !: !" > 0 (positive gradient) - For decreasing functions, , !: !" < 0 (negative gradient) (c) Connected rates of change. - Example: !< != = !< !> × !> != (from chain rule) (d) Maxima and minima. - We can use the first derivative test or the second derivative test to determine the natur