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MTH 102 MTH 102 Past Questions Objective Question
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Question

Differentiate the implicit functions x^2 sin y - e^(2y) cos x = 3x^2 + 2x - 5 with respect to x
Options
A dy/dx = (6x-2xsin y-sin x e^(2y)+2) / (x^2 cos y+2cos x e^(2y))
B dy/dx = (6x+2xsin y-sin x e^(2y)+2) / (x^2 cos y-2cos x e^(2y))
C dy/dx = (6x-2xsin y-sin x e^(2y)+2) / (x^2 cos y-2cos x e^(2y))
Correct Answer
D dy/dx = (-6x-2xsin y-sin x e^(2y)-2) / (x^2 cos y-2cos x e^(2y))
Correct Answer

Option C is the correct answer.

Detailed Explanation

Differentiate each term with respect to x, remembering that y is a function of x and applying the product and chain rules where necessary:d/dx [x2 sin y] ⇒ 2x sin y + x2 cos y (dy/dx)d/dx [e2y cos x] ⇒ (e2y * 2 dy/dx) cos x + e2y (-sin x) = 2e2y cos x (dy/dx) - e2y sin xd/dx [3x2 + 2x - 5] ⇒ 6x + 2Combine these derivatives into the equation:2x sin y + x2 cos y (dy/dx) - (2e2y cos x (dy/dx) - e2y sin x) = 6x + 22x sin y + x2 cos y (dy/dx) - 2e2y cos x (dy/dx) + e2y sin x = 6x + 2Group terms with dy/dx:dy/dx (x2 cos y - 2e2y cos x) = 6x + 2 - 2x sin y - e2y sin xFinally, solve for dy/dx:dy/dx = (6x + 2 - 2x sin y - e2y sin x) / (x2 cos y - 2e2y cos x).This matches option C when the terms in the numerator are rearranged.

Hint

Remember to apply the product rule and chain rule carefully for terms involving both x and y, and differentiate y implicitly (dy/dx).

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 553 of 930
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