MAT1512 Assignment 3 (ANSWERS) 2026

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Assessment
MAT1512 3
2026

, The A+Math Chicken

QUESTION 1
a)
Differentiate:
𝑦 = 𝑥 3 In𝑥

[4 marks]
ANSWER:

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


b)
Differentiate:
𝑦 = 𝑒 2𝑥 sin 𝑥

[4 marks]
ANSWER:

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


c)
Differentiate:
𝑥2 + 1
𝑦 = In ( )
𝑥

[4 marks]

, The A+Math Chicken

ANSWER:

𝑑𝑦 𝑑𝑦 𝑑𝑢
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 )
𝑥 𝑥(2𝑥) − (𝑥 2 + 1)(1)
= 2 ×[ ]
𝑥 +1 (𝑥)2
𝑥 2𝑥 2 − 𝑥 2 − 1
= 2 ×
𝑥 +1 𝑥2
𝑥 𝑥2 − 1
= 2 ×
𝑥 +1 𝑥2
𝑥2 − 1
=
𝑥(𝑥 2 + 1)


QUESTION 2
a)
Use implicit differentiation:
𝑥 2 + 𝑥𝑦 + 𝑦 2 = 7

𝑑𝑦
• Find
𝑑𝑥

[6 marks]
ANSWER:

𝑑𝑦 2 𝑑𝑦
(𝑥 + 𝑥𝑦 + 𝑦 2 ) = (7)
𝑑𝑥 𝑑𝑥
𝑑𝑦 2 𝑑𝑦 𝑑𝑦 2
(𝑥 ) + (𝑥𝑦) + (𝑦 ) = 0
𝑑𝑥 𝑑𝑥 𝑑𝑥
𝑑𝑦 𝑑𝑦
2𝑥 + [𝑥 ∙ (1 ∙ ) + 𝑦 ∙ 1] + 2𝑦 ∙ =0
𝑑𝑥 𝑑𝑥
𝑑𝑦 𝑑𝑦
2𝑥 + 𝑥 + 𝑦 + 2𝑦 =0
𝑑𝑥 𝑑𝑥
𝑑𝑦 𝑑𝑦
𝑥 + 2𝑦 = −2𝑥 − 𝑦
𝑑𝑥 𝑑𝑥