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The equation given for implicit differentiation is:

\[
\ln(x^2 + 5y) = \sin(xy)
\]

To find the derivative \( \frac{dy}{dx} \), we will differentiate both sides of the equation with respect to \(x\), applying the chain rule where necessary.

### Step-by-step solution:

#### Left-hand side:
\[
\frac{d}{dx} \left( \ln(x^2 + 5y) \right)
\]
Using the chain rule, this becomes:
\[
\frac{1}{x^2 + 5y} \cdot \left( 2x + 5 \frac{dy}{dx} \right)
\]
Here, \( 2x \) is the derivative of \( x^2 \), and \( 5 \frac{dy}{dx} \) is the derivative of \( 5y \) with respect to \(x\), where we applied the chain rule to differentiate \(y\) implicitly.

#### Right-hand side:
\[
\frac{d}{dx} \left( \sin(xy) \right)
\]
Using the product and chain rules, we get:
\[
\cos(xy) \cdot \left( y + x \frac{dy}{dx} \right)
\]
Here, \( y \) is the derivative of \( xy \) with respect to \( x \), and \( x \frac{dy}{dx} \) is the implicit differentiation of \(y\).

### Combining both sides:
Now, we have the equation:
\[
\frac{1}{x^2 + 5y} \cdot \left( 2x + 5 \frac{dy}{dx} \right) = \cos(xy) \cdot \left( y + x \frac{dy}{dx} \right)
\]

### Solving for \( \frac{dy}{dx} \):
To isolate \( \frac{dy}{dx} \), expand both sides and group all terms containing \( \frac{dy}{dx} \) on one side.

Would you like to continue with the full solution, or any specific part of this step detailed further?

Here are some questions related to this process:

1. Why do we apply the chain rule when differentiating \( \ln(x^2 + 5y) \)?
2. What is the role of implicit differentiation in handling \( y \) in the equation?
3. How does the product rule apply to the term \( \sin(xy) \)?
4. Could you solve for \( \frac{dy}{dx} \) explicitly by expanding both sides of the equation?
5. What would change in the process if the equation involved \( \cos(x^2 + y^2) \)?

**Tip:** When performing implicit differentiation, always treat \( y \) as a function of \( x \) and apply the chain rule when differentiating terms involving \( y \).

Learn how to perform implicit differentiation of the equation ln(x^2 + 5y) = sin(xy), using the chain rule and product rule. This step-by-step guide is designed for high school students in Grades 10-12, covering key mathematical concepts including logarithmic and trigonometric differentiation.

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