Differentiate implicit function: 2x^3 + 5xy + 6y^2 = 87

To differentiate the implicit function \( 2x^3 + 5xy + 6y^2 = 87 \) with respect to \( x \), we’ll use implicit differentiation. Since \( y \) is implicitly defined as a function of \( x \), we apply the chain rule to terms involving \( y \).

Given:
\[
2x^3 + 5xy + 6y^2 = 87
\]

### Step 1: Differentiate both sides with respect to \( x \)
Differentiating each term separately:

1. **Differentiate \( 2x^3 \):**
   \[
   \frac{d}{dx}(2x^3) = 6x^2
   \]

2. **Differentiate \( 5xy \):**
   Use the product rule, \(\frac{d}{dx}(5xy) = 5 \left( \frac{dy}{dx} \cdot x + y \cdot \frac{dx}{dx} \right) = 5 \left( y + x \frac{dy}{dx} \right)\).

3. **Differentiate \( 6y^2 \):**
   Using the chain rule,
   \[
   \frac{d}{dx}(6y^2) = 12y \cdot \frac{dy}{dx}
   \]

4. **Differentiate the constant \( 87 \):**
   \[
   \frac{d}{dx}(87) = 0
   \]

Putting these results together, we have:
\[
6x^2 + 5 \left( y + x \frac{dy}{dx} \right) + 12y \frac{dy}{dx} = 0
\]

### Step 2: Collect terms involving \(\frac{dy}{dx}\)
Expanding and rearranging:
\[
6x^2 + 5y + 5x \frac{dy}{dx} + 12y \frac{dy}{dx} = 0
\]

Combine terms with \(\frac{dy}{dx}\):
\[
6x^2 + 5y = - (5x + 12y) \frac{dy}{dx}
\]

### Step 3: Solve for \(\frac{dy}{dx}\)
Isolating \(\frac{dy}{dx}\), we get:
\[
\frac{dy}{dx} = \frac{- (6x^2 + 5y)}{5x + 12y}
\]

### Final Answer:
\[
\frac{dy}{dx} = \frac{-(6x^2 + 5y)}{5x + 12y}
\]

Would you like further details on any steps?

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### Related Questions:
1. How would you find the second derivative \( \frac{d^2y}{dx^2} \) of this implicit function?
2. What if the function were \( 2x^3 + 5xy + 6y^2 = c \), where \( c \) is a constant? How would \( c \) affect the derivative?
3. Can you show how to use implicit differentiation if \( y \) were a function of multiple variables?
4. How do we know that \( y \) can be written as a function of \( x \) locally around certain points?
5. How would the differentiation process change if one term were \( 2x^3y \) instead of \( 2x^3 \)?

#### Tip:
When using implicit differentiation, remember to apply the product and chain rules carefully on each term involving \( y \).

Learn how to find the derivative of the implicit function 2x^3 + 5xy + 6y^2 = 87 with respect to x using implicit differentiation. This problem involves the chain rule, product rule, and other essential calculus concepts. Suitable for students in Grades 11-12.

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