To find the derivative of the expression \( y \cdot y \) (which is \( y^2 \)) with respect to \( x \), we'll use implicit differentiation.

Given:

\[
y^2 = y \cdot y
\]

Differentiate both sides with respect to \( x \):

\[
\frac{d}{dx}(y^2) = \frac{d}{dx}(y \cdot y)
\]

Using the chain rule, the derivative of \( y^2 \) with respect to \( x \) is:

\[
\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}
\]

So, the derivative of \( y \cdot y \) (or \( y^2 \)) with respect to \( x \) is:

\[
\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}
\]

This result indicates that the derivative of \( y^2 \) with respect to \( x \) is twice the function \( y \) multiplied by the derivative of \( y \) with respect to \( x \) (i.e., \( 2y \cdot \frac{dy}{dx} \)).

Would you like any further details on this, or would you like to explore one of the related questions?

1. How do you apply implicit differentiation to other functions?
2. Can we find \(\frac{dy}{dx}\) if we have a specific function for \( y \)?
3. How does this result relate to finding slopes of curves?
4. How do you handle higher-order derivatives in similar expressions?
5. Can we simplify this expression further under certain conditions?

**Tip:** When differentiating products or powers of a function implicitly, always apply the chain rule to account for the variable dependencies.

Learn how to find the derivative of y^2 using implicit differentiation. This problem covers differentiation of products, implicit differentiation, and the application of the chain rule. Suitable for advanced high school students.

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