Find dy/dx for y2 + 3x + 2xy = 5x2

We will apply the differentiation rules:

• $\frac{d}{dx}(y^n) = ny^{n-1}\frac{dy}{dx}$ (chain rule)
• $\frac{d}{dx}(cx) = c$
• $\frac{d}{dx}(u \cdot v) = u'v + uv'$ (product rule)
• $\frac{d}{dx}(cx^n) = ncx^{n-1}$

Differentiating each term:

• For $y^2$: $\frac{d}{dx}(y^2) = 2y\frac{dy}{dx}$
• For $3x$: $\frac{d}{dx}(3x) = 3$
• For $2xy$: Use the product rule with $u=2x$ and $v=y$. $u' = \frac{d}{dx}(2x) = 2$ $v' = \frac{d}{dx}(y) = \frac{dy}{dx}$ So, $\frac{d}{dx}(2xy) = (2)(y) + (2x)\left(\frac{dy}{dx}\right) = 2y + 2x\frac{dy}{dx}$
• For $5x^2$: $\frac{d}{dx}(5x^2) = 5 \cdot 2x^{2-1} = 10x$

Subtract 3 and $2y$ from both sides:

$$2y\frac{dy}{dx} + 2x\frac{dy}{dx} = 10x - 3 - 2y$$

$$\frac{dy}{dx} = \frac{10x - 3 - 2y}{2y + 2x}$$

The derivative $\frac{dy}{dx}$ is:

$$\frac{10x - 3 - 2y}{2x + 2y}$$