Implicit Differentiation

What Is Implicit Differentiation?

Many equations mix $x$ and $y$ in ways that make isolating $y$ difficult or impossible—like $x^2 + y^2 = 25$ or $y^3 + xy = 7$. Implicit differentiation lets you find $\frac{dy}{dx}$ directly without rearranging. The core idea: treat $y$ as a function of $x$ and differentiate both sides of the equation, applying the chain rule to every $y$-term.

The Key Rule

Since $y$ is secretly a function of $x$, when you differentiate any power of $y$, the chain rule requires an extra factor:

$\frac{d}{dx}\left[y^n\right] = ny^{n-1}\cdot\frac{dy}{dx}$

This $\frac{dy}{dx}$ factor appears every time—forgetting it is the most common mistake.

The Four-Step Procedure

1. Differentiate both sides with respect to $x$. Apply the chain rule to all $y$-terms and the product/quotient rule wherever $x$ and $y$ multiply or divide.
2. Expand any products.
3. Collect all $\frac{dy}{dx}$ terms on one side; move everything else to the other.
4. Factor and solve for $\frac{dy}{dx}$.

Simple Example: The Circle

For $x^2 + y^2 = 25$:

$2x + 2y\frac{dy}{dx} = 0$

$\frac{dy}{dx} = -\frac{x}{y}$

Notice the answer depends on both $x$ and $y$—this is normal. Geometrically, $-x/y$ is the slope of the tangent line at any point on the circle.

Watch Out for Products

When $x$ and $y$ multiply, use the product rule first, then attach $\frac{dy}{dx}$ to the $y$ derivative:

$\frac{d}{dx}[xy] = y + x\frac{dy}{dx}$

Missing the $\frac{dy}{dx}$ on the second term is a frequent error. Always ask: "Is this a product of an $x$-part and a $y$-part?" If yes, product rule applies.