Differentiating $y$ involves finding the derivative of $y$ with respect to $x$, which measures how $y$ changes as $x$ changes. When $y$ is defined explicitly as a function of $x$ (e.g., $y = f(x)$), you can directly apply differentiation rules. However, in cases of implicit differentiation, $y$ is defined implicitly within an equation involving both $x$ and $y$.

Implicit Differentiation

1. Take the derivative of both sides of the equation with respect to $x$.
2. Use the chain rule for any term involving $y$, treating $y$ as a function of $x$. This means that when differentiating $y$, you multiply by $\frac{dy}{dx}$.
3. Solve for $\frac{dy}{dx}$.

Examples

1. Example 1: Circle Equation $x^2 + y^2 = r^2$ Differentiate both sides: $2x + 2y \frac{dy}{dx} = 0$ Rearranging gives: $\frac{dy}{dx} = -\frac{x}{y}$

2. Example 2: Ellipse Equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ Differentiate both sides: $\frac{2x}{a^2} + \frac{2y}{b^2} \frac{dy}{dx} = 0$ Rearranging gives: $\frac{dy}{dx} = -\frac{b^2}{a^2} \cdot \frac{x}{y}$

3. Example 3: Implicit Relation $x^3 + y^3 = 3xy$ Differentiate both sides: $3x^2 + 3y^2 \frac{dy}{dx} = 3\left(y + x \frac{dy}{dx}\right)$ Rearranging and solving for $\frac{dy}{dx}$ gives: $\frac{dy}{dx} = \frac{3x^2 - 3y}{3y^2 - 3x} = \frac{x^2 - y}{y^2 - x}$

Summary

Implicit differentiation allows you to find the derivative of $y$ when $y$ is not isolated. It is particularly useful for curves and relationships that are not functions in the traditional sense.

Would you like more examples or details on a specific concept? Here are some related questions:

1. How do you apply the chain rule in implicit differentiation?
2. Can you differentiate higher-order equations using implicit differentiation?
3. What are some common pitfalls in implicit differentiation?
4. How does implicit differentiation apply to real-world problems?
5. What is the geometric interpretation of implicit differentiation?

Tip: Always remember to include the derivative $\frac{dy}{dx}$ when differentiating terms involving $y$.