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MTH 102 MTH 102 Past Questions Objective Question
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Question

Obtain the derivative of the implicit function x2 siny - e2y cosx = 3x2 + 2x - 5 with respect to x.
Options
A <i>dy/dx</i> = (6<i>x</i> + 2<i>x</i>sin<i>y</i> - sin <i>x e</i><sup>2<i>y</i></sup> + 2) / (<i>x</i><sup>2</sup>cos<i>y</i> + 2cos<i>x e</i><sup>2<i>y</i></sup>)
B <i>dy/dx</i> = (6<i>x</i> + 2<i>x</i>sin<i>y</i> - sin <i>x e</i><sup>2<i>y</i></sup> + 2) / (<i>x</i><sup>2</sup>cos<i>y</i> - 2cos<i>x e</i><sup>2<i>y</i></sup>)
C <i>dy/dx</i> = (6<i>x</i> - 2<i>x</i>sin<i>y</i> - sin <i>x e</i><sup>2<i>y</i></sup> + 2) / (<i>x</i><sup>2</sup>cos<i>y</i> - 2cos<i>x e</i><sup>2<i>y</i></sup>)
Correct Answer
D <i>dy/dx</i> = (-6<i>x</i> - 2<i>x</i>sin<i>y</i> - sin <i>x e</i><sup>2<i>y</i></sup> - 2) / (<i>x</i><sup>2</sup>cos<i>y</i> - 2cos<i>x e</i><sup>2<i>y</i></sup>)
Correct Answer

Option C is the correct answer.

Detailed Explanation

To differentiate the implicit function x2 siny - e2y cosx = 3x2 + 2x - 5 with respect to x, we apply the chain rule and product rule where necessary:
  1. Differentiate x2 siny: d/dx(x2 siny) = 2x siny + x2 cosy (dy/dx)
  2. Differentiate -e2y cosx: d/dx(-e2y cosx) = -[ (e2y * 2 * dy/dx) cosx + e2y (-sinx) ] = -2e2y cosx (dy/dx) + e2y sinx
  3. Differentiate 3x2 + 2x - 5: d/dx(3x2 + 2x - 5) = 6x + 2
Combine these terms: 2x siny + x2 cosy (dy/dx) - 2e2y cosx (dy/dx) + e2y sinx = 6x + 2Group terms with dy/dx: (dy/dx) (x2 cosy - 2e2y cosx) = 6x + 2 - 2x siny - e2y sinxSolve for dy/dx: dy/dx = (6x + 2 - 2x siny - e2y sinx) / (x2 cosy - 2e2y cosx)This matches option C, where terms are reordered.

Hint

Remember to use the product rule for terms involving both x and y, and the chain rule for derivatives of y (e.g., d/dx(siny) = cosy dy/dx) and for composite functions like e2y.

Question Info

Course: MTH 102
Topic: MTH 102 Past Questions
Type: Objective
Difficulty: Not specified
Question 734 of 930
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