Implicit Differentiation — AP Calculus AB
1. What Is Implicit Differentiation? ★★☆☆☆ ⏱ 3 min
An explicit function is written in the form $y = f(x)$, where $y$ is explicitly isolated on one side of the equation. However, many mathematical relations (such as circles, ellipses, and more complex curves) cannot be easily or fully solved for $y$ in terms of $x$.
Implicit differentiation is a technique to find $ rac{dy}{dx}$ directly from the original implicit relation, without rearranging to isolate $y$. It is not a new differentiation rule—it is simply a systematic application of the chain rule to implicit functions of $x$. Because $y$ is treated as a function of $x$ even when not written explicitly, every time we differentiate a term containing $y$, we must multiply by $ rac{dy}{dx}$ by the chain rule.
2. The Core Implicit Differentiation Process ★★☆☆☆ ⏱ 4 min
The entire technique relies on one key chain rule result for any differentiable function $f(y)$, where $y$ is a function of $x$:
\frac{d}{dx}\left[f(y)\right] = f'(y) \cdot \frac{dy}{dx}
- Differentiate every term on both sides of the relation with respect to $x$
- Move all terms that include $ rac{dy}{dx}$ to the left side of the equation, and all other terms to the right
- Factor out $ rac{dy}{dx}$ from the left side
- Divide both sides by the remaining factor to isolate $ rac{dy}{dx}$ as a function of $x$ and $y$
When you have products or quotients of $x$ and $y$ terms (like $xy$, $x^2y$, or $ rac{y}{x}$), you still apply the product rule or quotient rule as normal, before adding the $ rac{dy}{dx}$ factor for $y$ terms.
Exam tip: Always apply product/quotient rule first for mixed $x$-$y$ terms, then add the $ rac{dy}{dx}$ factor. Skipping the product rule (e.g. omitting the $y$ term for $ rac{d}{dx}(xy)$) is the most common first mistake on AP exams.
3. Tangent and Normal Lines to Implicit Curves ★★★☆☆ ⏱ 3 min
One of the most common AP exam applications of implicit differentiation is finding the equation of a tangent or normal line to a point on an implicit curve. This works exactly the same way as finding tangent lines for explicit functions, once you have the slope $ rac{dy}{dx}$ from implicit differentiation.
- Confirm the given point $(x_0, y_0)$ lies on the original curve (AP sometimes tests this by giving a point not on the curve)
- Substitute $x_0$ and $y_0$ into your expression for $ rac{dy}{dx}$ to get the tangent slope $m_{\text{tan}}$
- The slope of the normal line (perpendicular to the tangent) is the negative reciprocal: $m_{\text{norm}} = -\frac{1}{m_{\text{tan}}}$
- Use point-slope form $y - y_0 = m(x - x_0)$ to write the final equation
Exam tip: Always read the question carefully: if it asks for a normal line, not a tangent, you must use the negative reciprocal slope. AP exam writers regularly test this to catch students who skim the question.
4. Second Derivatives of Implicit Functions ★★★☆☆ ⏱ 3 min
AP Calculus AB regularly asks for the second derivative $ rac{d^2y}{dx^2}$ of an implicit function, in terms of $x$ and $y$. The process is straightforward, but requires an extra step that many students forget.
After finding the first derivative $ rac{dy}{dx}$, you differentiate $ rac{dy}{dx}$ with respect to $x$ exactly as you differentiated the original equation: all terms containing $y$ or $ rac{dy}{dx}$ still require the chain rule, so you will get a $ rac{dy}{dx}$ factor when differentiating those terms. After differentiating, you must substitute the expression you already found for $ rac{dy}{dx}$ into the second derivative, so that the final result is only in terms of $x$ and $y$, not $ rac{dy}{dx}$. You can also use the original curve equation to simplify the final result by canceling constant terms.
Exam tip: Never leave $ rac{dy}{dx}$ in your final answer for the second derivative. AP graders will deduct points for unsubstituted $ rac{dy}{dx}$ terms on FRQ. Always substitute immediately after differentiating the first derivative.
5. AP Style Practice Problems ★★★★☆ ⏱ 3 min
Common Pitfalls
Why: Students focus so much on remembering the chain rule for $y$-terms that they forget mixed $x$-$y$ products require the product rule first
Why: Students get accustomed to differentiating $x$-terms and forget that every function of $y$ needs the chain rule factor
Why: Students think plugging in early simplifies the problem, but it leads to lost terms and messy algebra errors
Why: Students confuse horizontal and vertical tangent conditions
Why: Implicit relations have multiple $y$-values for one $x$, and the wrong $y$ gives the wrong slope