91Ó°ÊÓ

The chain rule is a fundamental concept in calculus used to differentiate composite functions. A composite function is one where you have one function inside another, like the expression \( e^{x^2y} \) in our given equation. To differentiate such expressions, the chain rule helps us by providing a systematic method:

• Differential outer function as if the inner function were a single variable.
• Multiply by the derivative of the inner function.

This approach is crucial when dealing with implicit differentiation, which is the process used to find the derivative when the dependent variable is not isolated. In this exercise, the chain rule was applied to differentiate the exponential term \( e^{x^2y} \). As a result, we got: \[ e^{x^2y} \cdot \left(x^2 \frac{dy}{dx} + 2xy\right) \] as part of the derivative, ensuring that both \( x \) and \( y \) dependencies were accounted for.

The derivative is a measure of how a function changes as its input changes. In simpler terms, it provides the rate of change or slope at any given point on a curve. Finding derivatives is a central task in any calculus problem, whether explicit or implicit. Here, the aim was to find \( \frac{dy}{dx} \), the derivative of \( y \) with respect to \( x \). In our original equation \( 2e^{x^2y} = x \), neither \( y \) nor \( x \) is isolated, meaning we must differentiate implicitly. Differentiating implicitly means:

• Treat \( y \) as a function of \( x \), even though it's not given explicitly as such.
• Apply differentiation to both sides of the equation, treating \( y \) as a function you have to take into account when differentiating with respect to \( x \).

Thus, the derivative of \( y \) was found by rearranging and solving the differentiated equation step by step.

Calculus problem solving involves several strategic steps, especially with implicit differentiation. When faced with such problems, here's a general strategy:

• Understand the problem: Read the problem carefully and identify what you are required to find, which, in this case, is \( \frac{dy}{dx} \).
• Setup the differentiation: Recognize parts of the problem that involve implicit differentiation and apply appropriate rules like the chain rule.
• Differentiate: Apply the rules methodically on both sides of the equation.
• Simplify and solve: After differentiating, simplify the equation and isolate \( \frac{dy}{dx} \).
• Substitute specific values if necessary: If the problem provides a specific point, substitute these values to find the derivative at that point.

In this exercise, finding \( \frac{dy}{dx} \) involved correctly applying these steps to determine its value as \( \frac{1}{8} \) at the point (2,0). Problem-solving in calculus often requires practice, patience, and following a logical sequence of steps.