*x*3(*x* + *y*) = *y*2(4*x* − *y*) Find *dy/dx* by implicit differentiation

To find \( \frac{dy}{dx} \) using implicit differentiation, let's start with the equation:

\[
x^3(x + y) = y^2(4x - y)
\]

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

Using the product rule, differentiate each side:

#### Left Side
\[
\frac{d}{dx} \left( x^3(x + y) \right) = x^3 \cdot \frac{d}{dx}(x + y) + (x + y) \cdot \frac{d}{dx}(x^3)
\]
The derivatives are:
\[
= x^3(1 + \frac{dy}{dx}) + (x + y) \cdot 3x^2
\]
Expanding:
\[
= x^3 + x^3 \frac{dy}{dx} + 3x^2(x + y)
\]

#### Right Side
\[
\frac{d}{dx} \left( y^2(4x - y) \right) = y^2 \cdot \frac{d}{dx}(4x - y) + (4x - y) \cdot \frac{d}{dx}(y^2)
\]
The derivatives are:
\[
= y^2(4 - \frac{dy}{dx}) + (4x - y) \cdot 2y \frac{dy}{dx}
\]
Expanding:
\[
= 4y^2 - y^2 \frac{dy}{dx} + 8xy \frac{dy}{dx} - 2y^2 \frac{dy}{dx}
\]

### Step 2: Combine Terms

Now, we have:

\[
x^3 + x^3 \frac{dy}{dx} + 3x^3 + 3x^2y = 4y^2 - y^2 \frac{dy}{dx} + 8xy \frac{dy}{dx} - 2y^2 \frac{dy}{dx}
\]

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

Combine

x^3(x + y) = y^2(4x - y). Find dy/dx by implicit differentiation.

Learn how to find dy/dx using implicit differentiation for the equation x^3(x + y) = y^2(4x - y). This solution covers key calculus concepts like the product rule and implicit differentiation, suitable for Grades 11-12.

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