Finding the Derivative dy/dx for sin y = x sin (a + y)

The given equation is $\sin y = x \sin (a + y)$. We need to find the derivative $\frac{dy}{dx}$. This involves implicit differentiation.

It is often easier to differentiate with respect to $y$ first if $x$ can be isolated. Let's rearrange the equation to express $x$ in terms of $y$ and the constant $a$:

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

Dividing both sides by $\sin (a + y)$ (assuming $\sin (a + y) \neq 0$), we get:

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

Now, we can differentiate $x$ with respect to $y$, i.e., find $\frac{dx}{dy}$. We will use the quotient rule for differentiation, which states that if $f(y) = \frac{u(y)}{v(y)}$, then $\frac{df}{dy} = \frac{v(y) \frac{du}{dy} - u(y) \frac{dv}{dy}}{(v(y))^2}$.

In our case, $u(y) = \sin y$ and $v(y) = \sin (a + y)$.

• The derivative of $u(y) = \sin y$ with respect to $y$ is $\frac{du}{dy} = \cos y$.
• The derivative of $v(y) = \sin (a + y)$ with respect to $y$ requires the chain rule. Let $z = a + y$. Then $\frac{dv}{dy} = \frac{d}{dz}(\sin z) \cdot \frac{dz}{dy} = \cos z \cdot \frac{d}{dy}(a + y) = \cos (a + y) \cdot (0 + 1) = \cos (a + y)$.

Applying the quotient rule:

$\frac{dx}{dy} = \frac{\sin (a + y) \cdot \frac{d}{dy}(\sin y) - \sin y \cdot \frac{d}{dy}(\sin (a + y))}{(\sin (a + y))^2}$

$\frac{dx}{dy} = \frac{\sin (a + y) \cos y - \sin y \cos (a + y)}{\sin^2 (a + y)}$

The numerator of this expression is in the form $\sin A \cos B - \cos A \sin B$, which is the expansion for $\sin (A - B)$. Here, $A = a + y$ and $B = y$.

So, the numerator is $\sin ((a + y) - y) = \sin a$.

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

We are asked to find $\frac{dy}{dx}$. We know that $\frac{dy}{dx} = \frac{1}{dx/dy}$ (provided $\frac{dx}{dy} \neq 0$).

$\frac{dy}{dx} = \frac{1}{\frac{\sin a}{\sin^2 (a + y)}}$

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

This matches one of the given options.

Revision Table: Key Differentiation Rules Used

Additional Information: Implicit Differentiation

Implicit differentiation is a technique used when a function $y$ cannot be easily expressed explicitly in terms of $x$, or when the relationship between $x$ and $y$ is given by an equation where $y$ is not isolated.

Steps for implicit differentiation to find $\frac{dy}{dx}$:

1. Differentiate both sides of the equation with respect to $x$. Remember that $y$ is a function of $x$, so apply the chain rule whenever differentiating a term involving $y$. For example, $\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$.
2. Collect all terms containing $\frac{dy}{dx}$ on one side of the equation and all other terms on the other side.
3. Factor out $\frac{dy}{dx}$ from the terms containing it.
4. Solve for $\frac{dy}{dx}$ by dividing by the factor multiplying $\frac{dy}{dx}$.

In this specific problem, rearranging the equation to solve for $x$ and finding $\frac{dx}{dy}$ first was a more direct route due to the structure of the equation, avoiding direct implicit differentiation of the original form with respect to $x$, which would involve product rule on $x \sin(a+y)$. However, both methods should yield the same result.