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The product rule is a crucial concept when dealing with derivatives, especially when the function involves a product of two or more variables. When differentiating a product of functions, you can't simply take the derivative of each function separately. Instead, the product rule states that the derivative of a product of two functions, say \( u(x) \) and \( v(x) \), is given by:

• \( \frac{d}{dx} [u(x)v(x)] = u(x)\frac{dv(x)}{dx} + v(x)\frac{du(x)}{dx} \)

This means you differentiate each function one at a time while keeping the other function intact, then add the results.
In our exercise, we apply the product rule to terms like \( 2xy^3 \) and \( -x^2y \). For example:

• Differentiate \( 2xy^3 \) using the product rule to get \( 2y^3 + 6xy^2(\frac{dy}{dx}) \).
• For \( -x^2y \), the product rule gives \( -2xy - x^2(\frac{dy}{dx}) \).

Notice how we take the derivative of \( y \) with respect to \( x \), which requires implicit differentiation.

The derivative is a measure of how a function changes as its input values change. Specifically, it represents the rate of change of a function with respect to a variable.
In mathematical terms, if you have a function \( y = f(x) \), the derivative \( \frac{dy}{dx} \) represents the slope of the tangent line to the curve of \( y \) at any point.
In the exercise, the goal is to find \( \frac{dy}{dx} \) for the equation:

• \( 2xy^3 - x^2y = 2 \)

Because there are terms with both \( x \) and \( y \), we need implicit differentiation. This method helps differentiate equations where \( y \) cannot be isolated easily. We find the derivative of each term and rearrange them to solve for \( \frac{dy}{dx} \).
Understanding derivatives helps in analyzing real-world dynamic systems, such as physical motion and economic growth.

The differentiation process is systematic, and it's essential to take it step-by-step to avoid mistakes. Here's how it unfolds in the given problem:
**Step 1:** Differentiate each term of the equation with respect to \( x \). Apply the product rule where needed.
**Step 2:** After differentiating both sides, gather all terms involving \( \frac{dy}{dx} \) onto one side of the equation and collect the remaining terms on the opposite side.

• This is crucial because you need to isolate \( \frac{dy}{dx} \) to solve for it.

**Step 3:** Notice that \( \frac{dy}{dx} \) is a common factor in some terms. Factor it out to simplify the equation.

• For example, from terms like \( 6xy^2(\frac{dy}{dx}) + x^2(\frac{dy}{dx}) \).

**Step 4:** Divide both sides of the equation by the factored expression to solve explicitly for \( \frac{dy}{dx} \).

• This will give the final expression for the derivative in terms of \( x \) and \( y \).

Following these organized steps ensures clarity in solving complex differentiation problems, aiding in both comprehension and execution.