Find 𝑑𝑦𝑑𝑥���� by implicit differentiation.

2𝑥^3𝑦^2−4𝑥^2𝑦=4




To find \(\frac{dy}{dx}\) by implicit differentiation, we'll differentiate both sides of the equation \( 2x^3y^2 - 4x^2y = 4 \) with respect to \(x\), while treating \(y\) as a function of \(x\) (i.e., using the chain rule when differentiating terms involving \(y\)).

### Step 1: Differentiate both sides
We differentiate each term:

\[
\frac{d}{dx}(2x^3y^2) - \frac{d}{dx}(4x^2y) = \frac{d}{dx}(4)
\]

#### First term: \(2x^3y^2\)
Using the product rule for differentiation: \(\frac{d}{dx}(uv) = u'v + uv'\), where \(u = 2x^3\) and \(v = y^2\), we get:

\[
\frac{d}{dx}(2x^3y^2) = 6x^2y^2 + 2x^3 \cdot 2y \cdot \frac{dy}{dx}
\]

Simplified:
\[
6x^2y^2 + 4x^3y \frac{dy}{dx}
\]

#### Second term: \(-4x^2y\)
Again, apply the product rule: \(u = -4x^2\) and \(v = y\), so:

\[
\frac{d}{dx}(-4x^2y) = -8xy - 4x^2 \cdot \frac{dy}{dx}
\]

#### Right-hand side: \(4\)
\[
\frac{d}{dx}(4) = 0
\]

### Step 2: Combine results
Putting everything together, we have:
\[
6x^2y^2 + 4x^3y \frac{dy}{dx} - 8xy - 4x^2 \frac{dy}{dx} = 0
\]

### Step 3: Solve for \(\frac{dy}{dx}\)
Group the terms involving \(\frac{dy}{dx}\) on one side:
\[
4x^3y \frac{dy}{dx} - 4x^2 \frac{dy}{dx} = -6x^2y^2 + 8xy
\]

Factor out \(\frac{dy}{dx}\) on the left side:
\[
\frac{dy}{dx}(4x^3y - 4x^2) = 8xy - 6x^2y^2
\]

Finally, solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{8xy - 6x^2y^2}{4x^3y - 4x^2}
\]

### Final Answer:
\[
\frac{dy}{dx} = \frac{8xy - 6x^2y^2}{4x^2(xy - 1)}
\]

Would you like more details on any step, or have any questions?

Here are 5 related questions you might be interested in:
1. How does implicit differentiation work?
2. What is the product rule in differentiation?
3. How does the chain rule apply when differentiating implicitly?
4. Can implicit differentiation be applied to trigonometric functions?
5. How do you solve implicit equations after differentiation?

**Tip:** When using implicit differentiation, always remember to apply the product rule and chain rule carefully to terms involving both \(x\) and \(y\).

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

Find dy/dx by applying implicit differentiation to the equation 2x^3y^2 − 4x^2y = 4. This problem involves using the product rule and chain rule to solve for dy/dx, providing a detailed step-by-step solution. Ideal for students in advanced high school calculus (Grades 11-12).

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