Implicit differentiation is a technique used to differentiate equations where the variables \( x \) and \( y \) are not explicitly separated, meaning \( y \) is not isolated on one side of the equation. Here's an example with a step-by-step explanation:

### Example: 
Differentiate the equation \( x^2 + y^2 = 25 \) implicitly with respect to \( x \).

### Step-by-step solution:

1. **Differentiate both sides with respect to \( x \)**:
   Since \( y \) is a function of \( x \), when differentiating any term involving \( y \), we apply the chain rule.

   \[
   \frac{d}{dx}(x^2 + y^2) = \frac{d}{dx}(25)
   \]

2. **Apply the derivative**:
   - The derivative of \( x^2 \) with respect to \( x \) is \( 2x \).
   - The derivative of \( y^2 \) with respect to \( x \) is \( 2y \frac{dy}{dx} \) by the chain rule (since \( y \) is a function of \( x \)).
   - The derivative of the constant \( 25 \) is \( 0 \).

   Thus, the equation becomes:
   \[
   2x + 2y \frac{dy}{dx} = 0
   \]

3. **Solve for \( \frac{dy}{dx} \)**:
   Isolate \( \frac{dy}{dx} \) by moving the terms around:

   \[
   2y \frac{dy}{dx} = -2x
   \]

   Divide both sides by \( 2y \):

   \[
   \frac{dy}{dx} = \frac{-x}{y}
   \]

### Final Answer:
The derivative \( \frac{dy}{dx} \) is \( \frac{-x}{y} \).

This shows how to implicitly differentiate when the variables are mixed in an equation.

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Would you like more details on any of the steps? Here are some related questions:

1. How do you apply the chain rule in implicit differentiation?
2. What if the equation involved higher powers of \( x \) and \( y \)?
3. Can implicit differentiation be used with trigonometric functions?
4. How do you apply implicit differentiation to find the second derivative?
5. What happens if you need to solve for \( y \) after differentiating?

**Tip:** When performing implicit differentiation, always apply the chain rule to terms involving \( y \) since \( y \) is treated as a function of \( x \).

Learn how to differentiate the equation x^2 + y^2 = 25 implicitly with respect to x using the chain rule. This step-by-step solution explains the key concepts of implicit differentiation, including how to apply the chain rule. Suitable for students in Grades 11-12.

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