Differentiate implicitly to find dy divided by dx. Then find the slope of the curve at the given point.
xsquaredysquaredequals36​,     ​(1​,6​)

To solve this, we will implicitly differentiate \( x^2 y^2 = 36 \) with respect to \( x \) to find \( \frac{dy}{dx} \). Then we will calculate the slope of the curve at the point \( (1, 6) \).

### Step 1: Implicit Differentiation
Starting with the equation:
\[
x^2 y^2 = 36
\]

1. Differentiate both sides with respect to \( x \):
   \[
   \frac{d}{dx}(x^2 y^2) = \frac{d}{dx}(36)
   \]

2. Applying the product rule on the left side, where \( u = x^2 \) and \( v = y^2 \):
   \[
   u'v + uv' = 0
   \]
   \[
   2x y^2 + x^2 \cdot 2y \cdot \frac{dy}{dx} = 0
   \]

3. Simplify:
   \[
   2x y^2 + 2x^2 y \frac{dy}{dx} = 0
   \]

4. Divide through by \( 2 \):
   \[
   x y^2 + x^2 y \frac{dy}{dx} = 0
   \]

5. Isolate \( \frac{dy}{dx} \):
   \[
   x^2 y \frac{dy}{dx} = -x y^2
   \]
   \[
   \frac{dy}{dx} = -\frac{x y^2}{x^2 y}
   \]

6. Simplify by canceling terms:
   \[
   \frac{dy}{dx} = -\frac{y}{x}
   \]

### Step 2: Find the Slope at \( (1, 6) \)

To find the slope at the point \( (1, 6) \), substitute \( x = 1 \) and \( y = 6 \) into \( \frac{dy}{dx} = -\frac{y}{x} \):

\[
\frac{dy}{dx} = -\frac{6}{1} = -6
\]

### Final Answer
The slope of the curve at the point \( (1, 6) \) is \( -6 \).

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#### Related Questions
1. How does implicit differentiation differ from explicit differentiation?
2. What is the product rule, and why is it used here?
3. How can the slope of a curve be found at multiple points?
4. What are the applications of implicit differentiation in calculus?
5. Why is it important to find \( \frac{dy}{dx} \) in terms of \( x \) and \( y \)?

#### Tip:
When implicitly differentiating, remember to apply the chain rule carefully whenever a variable depends on \( x \).

Differentiate implicitly to find dy divided by dx. Then find the slope of the curve at the given point.
 x^2 * y^2 = 36,     (1,6)

Explore how to use implicit differentiation to find dy/dx for the equation x^2 * y^2 = 36. Learn to apply the product and chain rules and determine the slope of the curve at the point (1,6). Suitable for advanced calculus students in Grade 12 or college.

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