Question Detail
Question
Differentiate the implicit functions \( x^2 \sin(y) - e^{2y} \cos(x) = 3x^2 + 2x - 5 \) with respect to \( x \)
Correct Answer
Option C is the correct answer.
Detailed Explanation
Differentiate each term with respect to ( x ):( \frac{d}{dx}(x^2 \sin y) - \frac{d}{dx}(e^{2y} \cos x) = \frac{d}{dx}(3x^2 + 2x - 5) )Using the product rule and chain rule:( (2x \sin y + x^2 \cos y \frac{dy}{dx}) - (e^{2y} (2\frac{dy}{dx}) \cos x + e^{2y} (-\sin x)) = 6x + 2 )( 2x \sin y + x^2 \cos y \frac{dy}{dx} - 2e^{2y} \cos x \frac{dy}{dx} + e^{2y} \sin x = 6x + 2 )Group terms with ( \frac{dy}{dx} ) on one side and other terms on the other side:( \frac{dy}{dx} (x^2 \cos y - 2e^{2y} \cos x) = 6x + 2 - 2x \sin y - e^{2y} \sin x )Solve for ( \frac{dy}{dx} ):( \frac{dy}{dx} = \frac{6x - 2x \sin y - e^{2y} \sin x + 2}{x^2 \cos y - 2e^{2y} \cos x} )This matches option C.
Hint
Apply the product rule and chain rule carefully. Remember that \( y \) is a function of \( x \), so \( \frac{d}{dx}(f(y)) = f'(y) \frac{dy}{dx} \).