If $x \sin (a+y) = \sin Y$, Find $\frac{dy}{dx}$.

In this article, we will delve into the problem of finding the derivative $\frac{dy}{dx}$ given the equation $x \sin (a+y) = \sin y$. This problem is a classic example of implicit differentiation, a technique used when it's difficult or impossible to express $y$ explicitly as a function of $x$. We'll explore the step-by-step process of differentiating both sides of the equation with respect to $x$, applying the chain rule, and then isolating $\frac{dy}{dx}$ to obtain the final result. This exploration will not only provide a solution to the specific problem but also enhance your understanding of implicit differentiation, a fundamental concept in calculus.

Problem Statement

Given the equation:

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

Find $\frac{dy}{dx}$.

Solution

To find $\frac{dy}{dx}$, we will use the method of implicit differentiation. Implicit differentiation is a powerful technique used when it is difficult or impossible to isolate $y$ as a function of $x$. In this method, we differentiate both sides of the equation with respect to $x$, treating $y$ as a function of $x$. This involves applying the chain rule whenever we differentiate a term involving $y$.

Step 1: Differentiate both sides with respect to $x$

We start by differentiating both sides of the given equation with respect to $x$:

$\frac{d}{dx} [x \sin (a+y)] = \frac{d}{dx} [\sin y]$

Step 2: Apply the product rule and chain rule

On the left side, we have a product of two functions of $x$, namely $x$ and $\sin(a+y)$. We apply the product rule, which states that $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$. On the right side, we apply the chain rule since we are differentiating $\sin y$ with respect to $x$, where $y$ is a function of $x$.

Applying the product rule on the left side:

$\frac{d}{dx} [x \sin (a+y)] = x \frac{d}{dx} [\sin (a+y)] + \sin (a+y) \frac{d}{dx} [x]$

Applying the chain rule on the right side:

$\frac{d}{dx} [\sin y] = \cos y \frac{dy}{dx}$

Now, let's compute the derivatives:

• $\frac{d}{dx} [x] = 1$
• $\frac{d}{dx} [\sin (a+y)] = \cos (a+y) \frac{d}{dx} (a+y) = \cos (a+y) (0 + \frac{dy}{dx}) = \cos (a+y) \frac{dy}{dx}$

Substituting these results into the differentiated equation:

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

Step 3: Rearrange the equation to isolate $\frac{dy}{dx}$

Now we need to isolate $\frac{dy}{dx}$. First, let's gather all terms containing $\frac{dy}{dx}$ on one side of the equation and the remaining terms on the other side:

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

Factor out $\frac{dy}{dx}$ from the left side:

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

Now, divide both sides by $[x \cos (a+y) - \cos y]$ to solve for $\frac{dy}{dx}$:

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

Step 4: Simplify the expression for $\frac{dy}{dx}$

To simplify the expression, we can use the original equation $x \sin (a+y) = \sin y$. We can express $x$ in terms of $y$ and $a$:

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

Substitute this expression for $x$ into the equation for $\frac{dy}{dx}$:

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

Multiply the numerator and denominator by $\sin (a+y)$ to eliminate the fraction in the denominator:

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

Now, we can use the sine subtraction formula: $\sin(A - B) = \sin A \cos B - \cos A \sin B$. In our case, $A = y$ and $B = a+y$, so:

$\sin y \cos (a+y) - \cos y \sin (a+y) = \sin [y - (a+y)] = \sin (y - a - y) = \sin (-a) = -\sin a$

Substitute this back into the expression for $\frac{dy}{dx}$:

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

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

Final Answer

Therefore, the derivative $\frac{dy}{dx}$ is:

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

So, the correct answer is option B.

In this comprehensive solution, we successfully found the derivative $\frac{dy}{dx}$ for the given implicit equation $x \sin (a+y) = \sin y$. We meticulously applied the principles of implicit differentiation, including the product rule and chain rule, to differentiate both sides of the equation with respect to $x$. The process involved careful algebraic manipulation to isolate $\frac{dy}{dx}$, followed by simplification using trigonometric identities and the original equation. The final result, $\frac{dy}{dx} = \frac{\sin^2 (a+y)}{\sin a}$, not only solves the specific problem but also provides a valuable illustration of the power and elegance of implicit differentiation in calculus. Mastering these techniques is crucial for tackling a wide range of problems in calculus and related fields.

This detailed walkthrough highlights the importance of understanding fundamental calculus concepts and their application in solving complex problems. By breaking down the problem into manageable steps and providing clear explanations, we have demonstrated how to approach and solve implicit differentiation problems effectively. This skill is essential for anyone studying calculus and its applications in various scientific and engineering disciplines.