For these questions:

Question 1

The best answer for why we might need to use implicit differentiation is:

• The equation giving the relationship between the variable is NOT solved for the dependent variable.

Explanation: Implicit differentiation is often used when we have an equation that defines a relationship between variables, but it’s difficult or impossible to explicitly solve for one variable in terms of the other. By differentiating both sides of the equation with respect to the independent variable, we can find the derivative even without an explicit function.

Question 2

To differentiate $y^2 = (f(x))^2$ with respect to $x$, we apply the chain rule.

1. For $\frac{d}{dx}(y^2)$: $\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx} = 2y \cdot f'(x) \quad \text{(since } y = f(x) \text{)}$

2. This means the correct answers are:

   • $2y \cdot \frac{dy}{dx}$
   • $2f(x) \cdot f'(x)$

Thus, the selected answers are:

• 2y(dy/dx)*
• 2f(x)f'(x)

Would you like a deeper explanation of the chain rule application or any of these concepts?

Follow-up Questions

1. How is implicit differentiation used in related rates problems?
2. What are the typical scenarios where explicit differentiation is preferred over implicit?
3. How would the process change if $y$ were expressed explicitly in terms of $x$?
4. How does the chain rule apply in the differentiation of composite functions?
5. Why is $\frac{d}{dx}(y^2)$ not simply $2y$?

Tip

Remember, when differentiating an expression involving a function of $x$, always apply the chain rule if that function is itself not isolated or has multiple terms squared.