Implicit differentiation is a technique used to differentiate equations where \( y \) is not explicitly solved for in terms of \( x \). This is useful when dealing with relations between variables that cannot easily be rewritten as \( y = f(x) \).

Here’s a step-by-step guide on how to perform implicit differentiation:

### Steps for Implicit Differentiation:

1. **Differentiate both sides of the equation with respect to \( x \)**: Treat \( y \) as a function of \( x \) (i.e., \( y = y(x) \)) and apply the chain rule to differentiate terms involving \( y \). When differentiating a term with \( y \), multiply by \( \frac{dy}{dx} \) (often written as \( y' \)).

2. **Apply the product rule and chain rule where necessary**: If the equation involves products or compositions of functions (like \( xy \)), apply the product or chain rule accordingly.

3. **Collect all terms involving \( \frac{dy}{dx} \) on one side of the equation**.

4. **Solve for \( \frac{dy}{dx} \)** to find the derivative.

### Example:
Consider the equation \( x^2 + y^2 = 25 \).

1. Differentiate both sides with respect to \( x \):
   \[
   \frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(25)
   \]
   The derivative of \( x^2 \) is \( 2x \), and using the chain rule, the derivative of \( y^2 \) is \( 2y \cdot \frac{dy}{dx} \) (since \( y \) is a function of \( x \)).
   \[
   2x + 2y \cdot \frac{dy}{dx} = 0
   \]

2. Solve for \( \frac{dy}{dx} \):
   \[
   2y \cdot \frac{dy}{dx} = -2x
   \]
   \[
   \frac{dy}{dx} = \frac{-x}{y}
   \]

Thus, the implicit derivative is \( \frac{dy}{dx} = \frac{-x}{y} \).

Would you like more details on any specific part of this process or another example?

### Relative questions:
1. How is implicit differentiation different from explicit differentiation?
2. What role does the chain rule play in implicit differentiation?
3. Can implicit differentiation be used for higher-order derivatives?
4. How would you apply implicit differentiation to trigonometric equations?
5. What is the significance of solving for \( \frac{dy}{dx} \) in real-world applications?

### Tip:
Always remember to apply the chain rule carefully when differentiating terms involving \( y \) in implicit differentiation.

Understand how to perform implicit differentiation to find the derivative of equations where y is not explicitly solved in terms of x. This guide explains the chain rule, product rule, and includes a step-by-step example of differentiating the equation x^2 + y^2 = 25. Ideal for students in Grades 10-12.

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