x^2(x + y) = y^2(4x − y) find dy/dx by implicit differentiation

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

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

we will differentiate both sides of the equation with respect to \( x \).

### Step 1: Differentiate Both Sides
The equation is:

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

Differentiating both sides with respect to \( x \):

#### Left Side
For the left side, we have \( x^2(x + y) \). We can apply the product rule here.

\[
\frac{d}{dx} \left( x^2(x + y) \right) = \frac{d}{dx}(x^2) \cdot (x + y) + x^2 \cdot \frac{d}{dx}(x + y)
\]

1. \( \frac{d}{dx}(x^2) = 2x \), so \( 2x(x + y

Learn how to find dy/dx for the equation x^2(x + y) = y^2(4x − y) using implicit differentiation. This problem demonstrates key calculus concepts such as the product rule and chain rule, with detailed steps suitable for advanced students in calculus.

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