Find 𝑑𝑦/𝑑𝑥
 by implicit differentiation.

4𝑥^3𝑦^2−10𝑥^2𝑦=6

To find \(\frac{dy}{dx}\) using implicit differentiation for the equation 

\[
4x^3y^2 - 10x^2y = 6,
\]

follow these steps:

1. Differentiate both sides with respect to \(x\).

   - The derivative of \(4x^3y^2\) requires the product rule:
     \[
     \frac{d}{dx}(4x^3y^2) = 4(3x^2)y^2 + 4x^3(2y\frac{dy}{dx}) = 12x^2y^2 + 8x^3y\frac{dy}{dx}.
     \]
   
   - The derivative of \(-10x^2y\) also requires the product rule:
     \[
     \frac{d}{dx}(-10x^2y) = -10(2x)y - 10x^2\frac{dy}{dx} = -20xy - 10x^2\frac{dy}{dx}.
     \]

   - The derivative of \(6\) is \(0\).

2. Combine the derivatives:
   \[
   12x^2y^2 + 8x^3y\frac{dy}{dx} - 20xy - 10x^2\frac{dy}{dx} = 0.
   \]

3. Rearrange the equation to isolate \(\frac{dy}{dx}\):
   \[
   (8x^3y - 10x^2)\frac{dy}{dx} = -12x^2y^2 + 20xy.
   \]

4. Solve for \(\frac{dy}{dx}\):
   \[
   \frac{dy}{dx} = \frac{-12x^2y^2 + 20xy}{8x^3y - 10x^2}.
   \]

Thus, the final result is:

\[
\frac{dy}{dx} = \frac{20xy - 12x^2y^2}{8x^3y - 10x^2}.
\]

If you would like further details or have any questions, feel free to ask! Here are some related questions to consider:

1. What is the geometric interpretation of implicit differentiation?
2. Can you provide a specific example of where implicit differentiation is useful?
3. How would you solve for \(y\) explicitly from the given equation?
4. What happens to the derivative if you change the equation slightly?
5. Are there other methods to find derivatives of implicitly defined functions? 

**Tip:** When applying implicit differentiation, always remember to differentiate each term carefully, paying attention to the product rule for terms involving both \(x\) and \(y\).

Find dy/dx by implicit differentiation. 4x^3y^2−10x^2y=6

Explore the steps to find dy/dx using implicit differentiation for the equation 4x^3y^2 - 10x^2y = 6. This guide covers key calculus concepts, including the product rule and the implicit function theorem, tailored for students in Grades 11-12.

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