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The chain rule is a fundamental technique in calculus for finding the derivative of a composition of two or more functions. Implicit differentiation, such as in our example, often relies on the chain rule. When you have a variable, like y, which itself is a function of x, any time you differentiate a term with y in it, you apply the chain rule.

Here's the core idea: if you have a function within another function—say u(y) within f(u)—and you want to find the derivative of f with respect to x, you first find the derivative of f with respect to u, and then multiply it by the derivative of u with respect to x. This can be represented as \( \frac{df}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \).

For instance, when differentiating \(7y^3\) with respect to x, you treat y as a function of x (let's call it u), and apply the chain rule: \( \frac{d}{dx}(7y^3) = 7 \cdot 3y^2 \cdot \frac{dy}{dx} \), where \( \frac{dy}{dx} \) is the derivative of y with respect to x. It's essential to understand this rule to differentiate correctly in implicit differentiation problems.

When you differentiate an equation with respect to x, you're finding out how the y-part of the equation changes as x changes. This is like asking, 'If I move a bit along the x-axis, how does the y-value wiggle around?' When differentiating explicit functions where y is expressed as a function of x, you use the standard differentiation rules. But, in cases of implicit differentiation—where y is not isolated—every time you differentiate a y term, you multiply by \(\frac{dy}{dx}\) because y is implicitly a function of x.

As seen in our exercise, you apply this approach to each term individually: differentiate \(6x^3\) as you usually would for an explicit x function, but handle the term \(13xy\) by differentiating both x and y parts, remembering that y is a 'hidden' function of x. This careful attention ensures that all y changes with respect to x are accounted for.

Once you have applied the chain rule and differentiated all terms with respect to x, the next step is solving for \(\frac{dy}{dx}\). This derivative represents the slope of the tangent line to a curve at any point - a crucial part of understanding the rate at which y changes as x changes. To isolate \(\frac{dy}{dx}\), you collect all terms including \(\frac{dy}{dx}\) on one side of the equation and move other terms to the opposite side as shown in our step-by-step solution.

After collecting the terms, you factor out \(\frac{dy}{dx}\) to simplify the equation. You then divide both sides by the remaining factor to solve for \(\frac{dy}{dx}\). The result gives you the relationship between x and y's rates of change. In practice, this process is essential for analyzing curves and understanding how variables affect each other—not just in mathematics but also in real-world applications such as physics and engineering.