Q.If $x^2+y^2=25$, find $\dfrac{dy}{dx}$ using implicit differentiation.

This equation describes a circle and is not written with $y$ isolated, so we differentiate both sides with respect to $x$ directly, treating $y$ as an (unspecified) function of $x$ and applying the chain rule to the $y^2$ term.

Step 1 — Differentiate both sides with respect to $x$. The left side has two terms: $x^2$ and $y^2$. The right side, $25$, is a constant.

• $\dfrac{d}{dx}(x^2) = 2x$ (ordinary power rule, since $x$ is the variable we differentiate with respect to).
• $\dfrac{d}{dx}(y^2) = 2y\,\dfrac{dy}{dx}$ — this uses the chain rule, because $y$ is itself a function of $x$, so differentiating $y^2$ requires the extra factor $\dfrac{dy}{dx}$.
• $\dfrac{d}{dx}(25) = 0$.

Step 2 — Assemble the differentiated equation.

Step 3 — Solve algebraically for $\dfrac{dy}{dx}$.

$2y\,\frac{dy}{dx} = -2x \quad\Rightarrow\quad \frac{dy}{dx} = -\frac{2x}{2y} = -\frac{x}{y}$ …

Alternative: solve for $y$ explicitly as $y=\pm\sqrt{25-x^2}$ and differentiate directly using the chain rule, as shown in the Check step. This works here because the equation happens to be solvable for $y$; implicit d …

The single most common error is differentiating $y^2$ as $2y$ (forgetting the chain-rule factor $\dfrac{dy}{dx}$), which is only correct if $y$ were the independent variable — since we are differentiatin …