The equation provided is:

$y^2 = x^2 + \sin(xy).$

To solve this using implicit differentiation, let's proceed step by step:

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

We apply the derivative to each term, remembering to treat $y$ as a function of $x$ (i.e., $y = y(x)$), which requires the chain rule:

$\frac{d}{dx}\big(y^2\big) = \frac{d}{dx}\big(x^2\big) + \frac{d}{dx}\big(\sin(xy)\big).$

Step 2: Apply the chain rule term-by-term

1. Derivative of $y^2$: $\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}.$

2. Derivative of $x^2$: $\frac{d}{dx}(x^2) = 2x.$

3. Derivative of $\sin(xy)$:

   • First, recall that $\sin(xy)$ involves the product $xy$, so use the product rule: $\frac{d}{dx}\big(\sin(xy)\big) = \cos(xy) \cdot \frac{d}{dx}(xy).$
   • Now apply the product rule to $xy$: $\frac{d}{dx}(xy) = x \frac{dy}{dx} + y.$
   • Thus: $\frac{d}{dx}\big(\sin(xy)\big) = \cos(xy) \cdot \big(x \frac{dy}{dx} + y\big).$

Step 3: Combine the results

Substitute all the differentiated terms back into the original equation:

$2y \frac{dy}{dx} = 2x + \cos(xy) \cdot \big(x \frac{dy}{dx} + y\big).$

Step 4: Solve for $\frac{dy}{dx}$

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

1. Expand the terms involving $\cos(xy)$: $2y \frac{dy}{dx} = 2x + \cos(xy) \cdot x \frac{dy}{dx} + \cos(xy) \cdot y.$

2. Collect all terms involving $\frac{dy}{dx}$ on one side: $2y \frac{dy}{dx} - \cos(xy) \cdot x \frac{dy}{dx} = 2x + \cos(xy) \cdot y.$

3. Factor out $\frac{dy}{dx}$ on the left-hand side: $\frac{dy}{dx} \big(2y - \cos(xy) \cdot x\big) = 2x + \cos(xy) \cdot y.$

4. Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{2x + \cos(xy) \cdot y}{2y - \cos(xy) \cdot x}.$

Final Answer:

$\boxed{\frac{dy}{dx} = \frac{2x + \cos(xy) \cdot y}{2y - \cos(xy) \cdot x}.}$

Let me know if you want further clarification or additional steps explained!

Related Questions:

1. How do you apply the chain rule to composite functions like $\sin(xy)$?
2. What is the difference between implicit and explicit differentiation?
3. How do you verify if a solution obtained by implicit differentiation is correct?
4. What is the geometric interpretation of $\frac{dy}{dx}$ in this case?
5. Can you solve this problem explicitly for $y$ in terms of $x$ before differentiating?

Tip:

When using implicit differentiation, always carefully apply the chain rule, especially when dealing with products or trigonometric functions involving $y$.