Question:

\(\text{ If } \sin y = x \sin(a + y), \text{ then } \frac{dy}{dx} \text{ is:}\)

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When differentiating an implicit equation, always ensure to apply the chain rule correctly. In cases where the equation involves trigonometric functions, be sure to use their derivative identities. Additionally, when isolating \(\frac{dy}{dx}\), carefully rearrange terms to simplify the expression. In this problem, substituting the expression for \(x\) into the denominator helped simplify the final result.

Updated On: Mar 28, 2025
  • \( \frac{\sin \frac{a}{2}}{\sin(a + y)} \)
  • \( \frac{\sin(a + y)}{\sin^2 a} \)
  • \( \frac{\sin(a + y)}{\sin a} \)
  • \( \frac{\sin^2(a + y)}{\sin a} \)
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The Correct Option is D

Approach Solution - 1

The given equation is:

\(\sin y = x \sin(a + y)\).

Differentiate both sides with respect to \(x\):

\(\cos y \frac{dy}{dx} = \sin(a + y) + x \cos(a + y) \frac{dy}{dx}\).

Rearrange to isolate \(\frac{dy}{dx}\):

\(\frac{dy}{dx} (\cos y - x \cos(a + y)) = \sin(a + y)\).

Simplify:

\(\frac{dy}{dx} = \frac{\sin(a + y)}{\cos y - x \cos(a + y)}\).

From the original equation \(\sin y = x \sin(a + y)\), rewrite \(x\) as:

\(x = \frac{\sin y}{\sin(a + y)}\).

Substitute \(x\) into the denominator:

\(\cos y - x \cos(a + y) = \cos y - \frac{\sin y \cos(a + y)}{\sin(a + y)}\).

Simplify the denominator:

\(\cos y - x \cos(a + y) = \frac{\cos y \sin(a + y) - \sin y \cos(a + y)}{\sin(a + y)} = \frac{\sin a}{\sin(a + y)}\).

Substitute this back into \(\frac{dy}{dx}\):

\(\frac{dy}{dx} = \frac{\sin(a + y)}{\frac{\sin a}{\sin(a + y)}} = \frac{\sin^2(a + y)}{\sin a}\).

Thus:

\(\boxed{\frac{\sin^2(a + y)}{\sin a}}\).

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Approach Solution -2

The given equation is:

\(\sin y = x \sin(a + y)\).

Differentiate both sides with respect to \(x\):

\[ \cos y \frac{dy}{dx} = \sin(a + y) + x \cos(a + y) \frac{dy}{dx}. \]

Rearrange to isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} (\cos y - x \cos(a + y)) = \sin(a + y). \]

Simplify:

\[ \frac{dy}{dx} = \frac{\sin(a + y)}{\cos y - x \cos(a + y)}. \]

From the original equation \(\sin y = x \sin(a + y)\), rewrite \(x\) as:

\[ x = \frac{\sin y}{\sin(a + y)}. \]

Substitute \(x\) into the denominator:

\[ \cos y - x \cos(a + y) = \cos y - \frac{\sin y \cos(a + y)}{\sin(a + y)}. \]

Simplify the denominator:

\[ \cos y - x \cos(a + y) = \frac{\cos y \sin(a + y) - \sin y \cos(a + y)}{\sin(a + y)} = \frac{\sin a}{\sin(a + y)}. \]

Substitute this back into \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \frac{\sin(a + y)}{\frac{\sin a}{\sin(a + y)}} = \frac{\sin^2(a + y)}{\sin a}. \]

Thus:

\[ \boxed{\frac{\sin^2(a + y)}{\sin a}}. \]
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