MAT1512 Assignment 3 2021

MAT1512 ASSIGNMENT 3 2021



Question 1



(a) 𝑓(𝑥) = ln(𝑥 + 𝑙𝑛𝑥)

Method 1 (Differentiate immediately)

1 𝑑
𝑓 ′ (𝑥) = × (𝑥 + 𝑙𝑛𝑥)
𝑥 + 𝑙𝑛𝑥 𝑑𝑥


1 1
𝑓 ′ (𝑥) = × (1 + )
𝑥 + 𝑙𝑛𝑥 𝑥


1 1 1
𝑓 ′ (𝑥) = ×( + )
𝑥 + 𝑙𝑛𝑥 1 𝑥


1 𝑥 1
𝑓 ′ (𝑥) = ×( + )
𝑥 + 𝑙𝑛𝑥 𝑥 𝑥

1 𝑥+1
𝑓 ′ (𝑥) = ×( )
𝑥 + 𝑙𝑛𝑥 𝑥


𝑥+1
𝑓 ′ (𝑥) =
𝑥(𝑥 + 𝑙𝑛𝑥)

𝑥+1
𝑓 ′ (𝑥) =
𝑥 2 + 𝑥𝑙𝑛𝑥


Method 2 (Implicit differentiation)

𝑓(𝑥) = ln(𝑥 + 𝑙𝑛𝑥)

𝑓 = ln(𝑥 + 𝑙𝑛𝑥)



𝑒 𝑓 = 𝑒 ln(𝑥+𝑙𝑛𝑥)

, 𝑒 𝑓 = 𝑥 + 𝑙𝑛𝑥


𝑑 𝑓 𝑑
𝑒 = (𝑥 + 𝑙𝑛𝑥)
𝑑𝑥 𝑑𝑥

𝑑𝑓 1
𝑒𝑓 × =1+
𝑑𝑥 𝑥

𝑑𝑓 1 1
= 𝑓 (1 + )
𝑑𝑥 𝑒 𝑥

𝑑𝑓 1 1
= ln(𝑥+𝑙𝑛𝑥) (1 + )
𝑑𝑥 𝑒 𝑥

𝑑𝑓 1 1
= (1 + )
𝑑𝑥 (𝑥 + 𝑙𝑛𝑥) 𝑥

1 1
𝑓 ′ (𝑥) = (1 + )
(𝑥 + 𝑙𝑛𝑥) 𝑥

1 𝑥 1
𝑓 ′ (𝑥) = ×( + )
(𝑥 + 𝑙𝑛𝑥) 𝑥 𝑥

1 𝑥+1
𝑓 ′ (𝑥) = ×( )
(𝑥 + 𝑙𝑛𝑥) 𝑥


𝑥+1
𝑓 ′ (𝑥) =
𝑥(𝑥 + 𝑙𝑛𝑥)

𝑥+1
𝑓 ′ (𝑥) =
𝑥2+ 𝑥𝑙𝑛𝑥


𝑥−1
(b) 𝑔(𝑥) = √𝑥 4 +1


Method 1 (Differentiate immediately)


𝑥−1
𝑔(𝑥) = √ 4
𝑥 +1


𝑥 − 1 1⁄2
𝑔(𝑥) = ( )
𝑥4 + 1