Q.If $xy+y^2=7$, find $\dfrac{dy}{dx}$ using implicit differentiation.

Here $y$ is entangled with $x$ inside a product term as well as a power term, so differentiating requires both the product rule (for $xy$) and the chain rule (for $y^2$) together.

Step 1 — Differentiate the $xy$ term using the product rule. Treating $x$ and $y$ as two functions of $x$ (with $x$ trivially its own derivative $1$): $\dfrac{d}{dx}(xy) = (1)(y) + x\left(\dfrac{dy}{dx}\right) = y+x\dfrac{dy}{dx}$.

Step 2 — Differentiate the $y^2$ term using the chain rule. $\dfrac{d}{dx}(y^2) = 2y\dfrac{dy}{dx}$.

Step 3 — Differentiate the constant right-hand side. $\dfrac{d}{dx}(7) = 0$.

Step 4 — Assemble the differentiated equation.

Step 5 — Collect all $\dfrac{dy}{dx}$ terms and solve.

$\frac{dy}{dx}(x+2y) = -y \quad\Rightarrow\quad \frac{dy}{dx} = -\frac{y}{x+2y}$ …

Applying the power rule alone to $xy$ (as if it were $x\cdot y$ with $y$ constant, giving just $y$) instead of the full product rule is the main error, since it drops the essential $x\dfrac{dy}{dx}$ term entirely. A second common slip is solving for $\dfrac{dy}{dx}$ incorrectly when collecting terms — forge …