Question

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

• A. \( \frac{\sin \frac{a}{2}}{\sin(a + y)} \)
• B. \( \frac{\sin(a + y)}{\sin^2 a} \)
• C. \( \frac{\sin(a + y)}{\sin a} \)
• D. \( \frac{\sin^2(a + y)}{\sin a} \)

Hint:

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.

Answer 1

The Correct Option is D

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:

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

Answer 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:

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