Differentiate implicit function: x^4 * y^6 = 15

To differentiate the implicit function \( x^4 \cdot y^6 = 15 \) with respect to \( x \), we use implicit differentiation, treating \( y \) as a function of \( x \) (i.e., \( y = y(x) \)).

Given:
\[
x^4 \cdot y^6 = 15
\]

### Step 1: Differentiate both sides with respect to \( x \)

Differentiating the left side using the product rule:
\[
\frac{d}{dx}(x^4 \cdot y^6) = \frac{d}{dx}(15)
\]

Since 15 is a constant, its derivative is 0:
\[
\frac{d}{dx}(x^4 \cdot y^6) = 0
\]

### Step 2: Apply the Product Rule

Using the product rule, \( \frac{d}{dx}(u \cdot v) = u'v + uv' \):
\[
\frac{d}{dx}(x^4) \cdot y^6 + x^4 \cdot \frac{d}{dx}(y^6) = 0
\]

#### Differentiate \( x^4 \) and \( y^6 \):

1. \( \frac{d}{dx}(x^4) = 4x^3 \).
2. To differentiate \( y^6 \), apply the chain rule:
   \[
   \frac{d}{dx}(y^6) = 6y^5 \cdot \frac{dy}{dx}.
   \]

Substitute these into the equation:
\[
4x^3 \cdot y^6 + x^4 \cdot 6y^5 \cdot \frac{dy}{dx} = 0.
\]

### Step 3: Solve for \( \frac{dy}{dx} \)

Rearrange to isolate \( \frac{dy}{dx} \):
\[
x^4 \cdot 6y^5 \cdot \frac{dy}{dx} = -4x^3 \cdot y^6.
\]

Now, divide both sides by \( x^4 \cdot 6y^5 \) to solve for \( \frac{dy}{dx} \):
\[
\frac{dy}{dx} = \frac{-4x^3 \cdot y^6}{6x^4 \cdot y^5}.
\]

### Step 4: Simplify

Cancel terms to simplify:
\[
\frac{dy}{dx} = \frac{-2y}{3x}.
\]

### Final Answer:
\[
\frac{dy}{dx} = -\frac{2y}{3x}.
\]

Would you like a more detailed explanation on any part of the process? 

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Here are five related questions you might find interesting:
1. How would you differentiate \( x^3 \cdot y^4 = 10 \) using implicit differentiation?
2. What if the equation was \( x^5 \cdot y^7 = 20 \); how would the derivative change?
3. How does implicit differentiation work for trigonometric functions, like \( \sin(xy) = x^2 + y^2 \)?
4. What are some real-world applications of implicit differentiation?
5. How would you find the second derivative of \( y \) with respect to \( x \) for \( x^4 \cdot y^6 = 15 \)?

**Tip:** In implicit differentiation, remember that every time you differentiate a term with \( y \), you need to multiply by \( \frac{dy}{dx} \) since \( y \) is a function of \( x \).

Explore the process of differentiating the implicit function x^4 * y^6 = 15 with respect to x. This solution includes the use of implicit differentiation, the product rule, and the chain rule. Suitable for advanced high school students in Grades 11-12.

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