Implicitly defined functions — AP Precalculus
1. What Are Implicitly Defined Functions? ★☆☆☆☆ ⏱ 3 min
An explicitly defined function writes the dependent output variable (usually $y$) directly as a function of the independent input variable (usually $x$), in the form $y = f(x)$, the standard form you have used for most of the course. An implicitly defined function is a relation between $x$ and $y$ written as a single equation $F(x,y) = 0$, where $y$ is treated as a function of $x$ even if we cannot solve for $y$ in terms of elementary functions.
2. Implicit Differentiation via the Chain Rule ★★☆☆☆ ⏱ 4 min
The core skill for working with implicit functions is implicit differentiation, which allows you to find $\frac{dy}{dx}$ without solving for $y$ explicitly. The key idea is that $y$ is a function of $x$, so any term involving $y$ is a composite function of $x$, which requires the chain rule when differentiating with respect to $x$.
\frac{d}{dx}\left[y^n\right] = n y^{n-1} \cdot \frac{dy}{dx}
Terms that only involve $x$ are differentiated normally, just like with explicit functions. After differentiating both sides of the implicit equation with respect to $x$, you rearrange terms to solve for $\frac{dy}{dx}$, which will usually be an expression in both $x$ and $y$ — this is expected and acceptable.
Exam tip: Always keep the $\frac{dy}{dx}$ factor when differentiating any term that includes $y$ — if you forget it, you will end up with an incorrect constant slope.
3. Evaluating $\frac{dy}{dx}$ at a Point on an Implicit Curve ★★★☆☆ ⏱ 3 min
Most AP Precalculus problems do not ask for the general form of $\frac{dy}{dx}$; instead, they ask for the slope of the tangent line at a given point that lies on the implicit curve. After you find the general expression for $\frac{dy}{dx}$ in terms of $x$ and $y$, you simply substitute the coordinates of the given point $(x_0, y_0)$ directly into the expression. A critical preliminary step most students skip is confirming the given point actually lies on the curve.
Exam tip: Always simplify the general derivative before substituting the point to reduce arithmetic errors — factoring out common constants early cuts down on miscalculations.
4. Finding the Equation of a Tangent Line to an Implicit Curve ★★★☆☆ ⏱ 4 min
One of the most common FRQ questions on this topic asks for the full equation of the tangent line to an implicit curve at a given point. This combines the skill of finding the slope via implicit differentiation with the point-slope form of a line you learned earlier in the course. The AP exam will usually accept either point-slope or slope-intercept form unless specified otherwise.
Exam tip: If the question asks for the tangent line, double-check that you did not mix up $x$ and $y$ when substituting into the derivative — swapping coordinates will give you the wrong slope.
5. AP-Style Concept Check ★★★★☆ ⏱ 3 min
Common Pitfalls
Why: Students are used to differentiating only terms with $x$, so they automatically treat $y$ as a constant instead of a function of $x$.
Why: Students remember to add the $\frac{dy}{dx}$ for the $y$ term but forget that the $x$ term also needs to be differentiated.
Why: Students are uncomfortable working with expressions in $x$ and $y$, so they force a solution for $y$ even when it's unnecessary.
Why: Students expect derivatives to be only in terms of $x$, so they unnecessarily substitute to eliminate $y$.
Why: Students assume the point given is on the curve and skip the check.