Find the derivative (dy)/(dx) of the curve x^2 + x^2y = 12.

Question

Find the derivative (dy)/(dx) of the curve x^2 + x^2y = 12.

Asked on March 28, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-03-28T03:15:34.836Z

Step 1: Find the derivative $\frac{dy}{dx}$ of the curve $x^2 + x^2y = 12$. The equation of the curve is $x^2 + x^2y = 12$. Differentiate both sides with respect to $x$: $$ \frac{d}{dx}(x^2) + \frac{d}{dx}(x^2y) = \frac{d}{dx}(12) $$ Using the product rule for $x^2y$: $$ 2x + (2xy + x^2\frac{dy}{dx}) = 0 $$ $$ 2x + 2xy + x^2\frac{dy}{dx} = 0 $$ Isolate

Accepted Answer · 2026-03-28T03:15:34.836Z

Step 1: Find the derivative $\frac{dy}{dx}$ of the curve $x^2 + x^2y = 12$. The equation of the curve is $x^2 + x^2y = 12$. Differentiate both sides with respect to $x$: $$ \frac{d}{dx}(x^2) + \frac{d}{dx}(x^2y) = \frac{d}{dx}(12) $$ Using the product rule for $x^2y$: $$ 2x + (2xy + x^2\frac{dy}{dx}) = 0 $$ $$ 2x + 2xy + x^2\frac{dy}{dx} = 0 $$ Isolate