2026 MAT1512 Assignment 3 (Solutions)

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MAT1512
2026 Assignment 3




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, QUESTION 1
a)
𝑑𝑦 𝑑 𝑑 3
= 𝑥3 ∙ (In𝑥) + In𝑥 ∙ (𝑥 )
𝑑𝑥 𝑑𝑥 𝑑𝑥
𝑑𝑦 1
= 𝑥 3 ∙ ( ) + In𝑥 ∙ (3𝑥 2 )
𝑑𝑥 𝑥
3
𝑑𝑦 𝑥
= + In𝑥 ∙ (3𝑥 2 )
𝑑𝑥 𝑥
𝑑𝑦
= 𝑥 2 + 3𝑥 2 In𝑥
𝑑𝑥
𝑑𝑦
∴ = 𝑥 2 (1 + 3In𝑥)
𝑑𝑥


b)
𝑑𝑦 𝑑 𝑑 2𝑥
= 𝑒 2𝑥 ∙ (sin 𝑥) + sin 𝑥 ∙ (𝑒 )
𝑑𝑥 𝑑𝑥 𝑑𝑥
𝑑𝑦
= 𝑒 2𝑥 ∙ cos 𝑥 + sin 𝑥 (𝑒 2𝑥 ∙ 2)
𝑑𝑥
𝑑𝑦
= 𝑒 2𝑥 cos 𝑥 + 2𝑒 2𝑥 𝑠𝑖𝑛 𝑥
𝑑𝑥
𝑑𝑦
∴ = 𝑒 2𝑥 (cos 𝑥 + 2𝑠𝑖𝑛 𝑥)
𝑑𝑥


𝑑𝑦 𝑑𝑦 𝑑𝑢
c) The CHAIN Rule: If 𝑦 = 𝑓(𝑢) , where 𝑢 = 𝑔(𝑥) then = 𝑑𝑢 × 𝑑𝑥
𝑑𝑥

𝑑𝑦 1 𝑑 𝑥2 + 1
= 2 × ( )
𝑑𝑥 𝑥 +1 𝑑𝑥 𝑥
( 𝑥 )
𝑑𝑦 𝑥 𝑥(2𝑥) − (𝑥 2 + 1)(1)
= ×[ ]
𝑑𝑥 𝑥 2 + 1 (𝑥)2
𝑑𝑦 𝑥 2𝑥 2 − 𝑥 2 − 1
= ×
𝑑𝑥 𝑥 2 + 1 𝑥2
2
𝑑𝑦 𝑥 𝑥 −1
= 2 ×
𝑑𝑥 𝑥 + 1 𝑥2
𝑑𝑦 𝑥2 − 1
∴ =
𝑑𝑥 𝑥(𝑥 2 + 1)