To find the derivative of \( y \) with respect to \( x \) (denoted as \( \frac{dy}{dx} \) or \( y' \)), we need to implicitly differentiate the given equation.

The equation is:

\[
4x^3 + y - 6xy = 0
\]

### Step 1: Differentiate both sides with respect to \( x \)
We will differentiate each term individually with respect to \( x \), keeping in mind that \( y \) is a function of \( x \), so we must apply the chain rule where necessary.

1. Differentiate \( 4x^3 \):
   \[
   \frac{d}{dx}(4x^3) = 12x^2
   \]

2. Differentiate \( y \) with respect to \( x \):
   \[
   \frac{d}{dx}(y) = y'
   \]

3. Differentiate \( -6xy \) with respect to \( x \). This requires the product rule:
   \[
   \frac{d}{dx}(-6xy) = -6 \left( x \cdot \frac{dy}{dx} + y \cdot \frac{d}{dx}(x) \right)
   = -6 \left( x \cdot y' + y \cdot 1 \right)
   = -6xy' - 6y
   \]

### Step 2: Substitute the derivatives back into the equation
Substitute the differentiated terms back into the original equation:

\[
12x^2 + y' - 6xy' - 6y = 0
\]

### Step 3: Solve for \( y' \)
To solve for \( y' \), first combine like terms:

\[
y' - 6xy' = 6y - 12x^2
\]

Factor out \( y' \) on the left side:

\[
y'(1 - 6x) = 6y - 12x^2
\]

Finally, solve for \( y' \):

\[
y' = \frac{6y - 12x^2}{1 - 6x}
\]

So the derivative of \( y \) with respect to \( x \) is:

\[
y' = \frac{6y - 12x^2}{1 - 6x}
\]

Would you like more details or have any questions? Here are 5 questions related to this problem:

1. How does the product rule apply when differentiating \( -6xy \)?
2. What is the significance of implicit differentiation in this problem?
3. Can you find the second derivative \( y'' \) from this result?
4. How does the chain rule help in differentiating terms like \( y \) with respect to \( x \)?
5. What are the possible critical points for the function \( y(x) \) based on this derivative?

**Tip:** Always remember to apply the chain rule when differentiating a function that depends on another variable.

Learn how to find the derivative dy/dx of y with respect to x from the equation 4x³ + y - 6xy = 0 using implicit differentiation. This problem covers advanced algebraic techniques including the product rule and chain rule. Suitable for advanced high school students and calculus learners.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation