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 nature of the stationary point. - Second derivative test: o If !: !" = 0 and !!: !"! < 0, then the stationary point is a maximum point. o If !: !" = 0 and !!: !"! > 0, then the stationary point is a minimum point.

4 Examples of First and Second Derivative tests Find the stationary point of the curve 𝑦 = 4𝑥) − 16𝑥 and determine the nature of the stationary points. (a) Use the first derivative test. 𝑦 = 4𝑥) − 16𝑥 !: !" = 8𝑥 − 16 At stationary point, !: !" = 0 8𝑥 − 16 = 0 𝑥 = 2 At 𝑥 = 2, 𝑦 = 4(2)) − 16(2) = −16 At (2, −16) x value 1.9 2 2.1 𝐝𝒚 𝐝𝒙 !: !" = 8(1.9) − 16 = −0.8 d𝑦 d𝑥 = 0 !: !" = 8(2.1) − 16 = 0.8 Shape It is a minimum point. (b) Use the second derivative test. 𝑦 = 4𝑥) − 16𝑥 !: !" = 8𝑥 − 16 !!: !"! = 8 > 0 for all values of 𝑥 Thus since !!: !"! > 0, the stationary point is a minimum point.

5 7. Differentiation of Composite Trigonometric Functions Function Example (a) ! !" [𝑘 sin 𝑓(𝑥)] = 𝑘[cos 𝑓(𝑥)][𝑓′(𝑥)] ! !" [2 sin(3𝑥 + 1)] = (3)(2) cos(3𝑥 + 1) = 6 cos(3𝑥 + 1) (b) ! !" [𝑘 cos 𝑓(𝑥)] = −𝑘[sin 𝑓(𝑥)][𝑓′(𝑥)] ! !" R % ) cos(𝑥) + 1)S = 5 % )6 [−sin(𝑥) + 1)](2𝑥) = −𝑥 sin(𝑥) + 1) (c) ! !" [𝑘 tan 𝑓(𝑥)] = 𝑘[sec# 𝑓(𝑥)][𝑓′(𝑥)] ! !" [3 tan(𝑥3 + 3𝑥)] = 3(3𝑥2 + 3) sec2(𝑥3 + 3𝑥) (d) ! !" [𝑘 sin& 𝑓(𝑥)] = 𝑘𝑛[sin'() 𝑓(𝑥)][cos 𝑓(𝑥)][𝑓′(𝑥)] (e) ! !" [𝑘 cos& 𝑓(𝑥)] = −𝑘𝑛[cos'() 𝑓(𝑥)][sin 𝑓(𝑥)][𝑓′(𝑥)] (f) ! !" [𝑘 tan& 𝑓(𝑥)] = 𝑘𝑛[tan&() 𝑓(𝑥)][sec# 𝑓(𝑥)][𝑓′(𝑥)] Examples of Composite Trigonometric Function (a) ! !" 6 ) # cos*(𝑥+ + 2𝑥)8 = ) # (4)[cos+(𝑥+ + 2𝑥)][− sin(𝑥+ + 2𝑥)](3𝑥# + 2) = −2(3𝑥# + 2) cos+(𝑥+ + 2𝑥) sin(𝑥+ + 2𝑥) (b) ! !" 63 tan# : ) # 𝑥# + 2;8 = 3(2) 6tan : ) # 𝑥# + 2;8 6sec# : ) # 𝑥# + 2;8 6 ) # (2𝑥)8 = 6𝑥 tan : ) # 𝑥# + 2; sec# : ) # 𝑥# + 2; 8. Differentiation of Composite Exponential and Logarithmic Functions Function Example (a) ! !" =𝑘𝑒,(")? = 𝑘𝑓′(𝑥) 𝑒,(") ! !" R2𝑒 1 3𝑥+2S = 2 5 % '6 𝑒 1 3𝑥+2 = ) ' 𝑒 1 3𝑥+2 (b) ! !" [ln 𝑓(𝑥)] = ,#(") ,(") ! !" [ln(3𝑥) + 𝑥)] = *"-% '"!-%

6 Differentiation – Practice Questions 1a 1. Differentiate the following with respect to x: (a) 2𝑥) + 5𝑥 + 7 Answer: (a) ___________________ (b) ? ) 𝑥& + ( '". + 𝑥 Answer: (b) ___________________ (c) 4𝑥) + ) '√" + 7 Answer: (c) ___________________

www.ApexEducators.com 7 (d) 5𝑎𝑥) + 3𝑏𝑥' + 5 Answer: (d) ___________________ (e) '"!$&"" "" Answer: (e) ___________________ (f) '√"$& √" Answer: (f) ___________________ (g) √𝑥 + % √" Answer: (g) ___________________

8 2. Find the gradient of the curve 𝑦 = ("$& "! at the point where the curve crosses the x-axis. Answer: gradient = ___________________ 3. The gradient of the curve at 𝑦 = @ "! + A " at the point (−1, 5) is 4. Find the values of a and b. Answer: a = ___________________ b = ___________________

9 4. Differentiate the following with respect to x: (a) (2𝑥 + 5)? Answer: (a) ___________________ (b) 3(𝑥 + 4)( Answer: (b) ___________________ (c) ) ' 5 " * − 16 & Answer: (c) ___________________

10 (d) % '"-) Answer: (d) ___________________ (e) %) )-'"! Answer: (e) ___________________ (f) √5𝑥) + 6 Answer: (f) ___________________