Behaviors of implicit relations

Q: What is an implicit relation in the context of calculus?

An implicit relation is a relationship between variables, such as \(x^2 + y^2 = 25\), where \(y\) is not isolated as a function of \(x\) and may not pass the vertical line test.

Q: Under what condition does an implicit relation have a horizontal tangent line?

A horizontal tangent occurs when the numerator of the derivative \(\frac{dy}{dx}\) is zero, provided the denominator is not zero.

Q: Under what condition does an implicit relation have a vertical tangent line?

A vertical tangent occurs when the denominator of the derivative \(\frac{dy}{dx}\) is zero, provided the numerator is not zero.

Q: What does it mean if both the numerator and denominator of \(\frac{dy}{dx}\) are zero at a specific point?

If both are zero (the \(\frac{0}{0}\) case), the behavior is indeterminate, often indicating a sharp point (cusp) or where the curve crosses itself.

Q: How do you determine if an implicit relation is increasing or decreasing at a specific point \((x, y)\)?

Plug both the \(x\) and \(y\) coordinates into \(\frac{dy}{dx}\); if the result is positive, the relation is increasing; if negative, it is decreasing.

Q: What is the "Crucial Step" when finding the second derivative of an implicit relation?

Whenever you differentiate a term containing \(y\) with respect to \(x\), you must multiply by \(\frac{dy}{dx}\) due to the Chain Rule.

Q: Why must you perform a "Substitution Step" when finding the second derivative expression?

You must replace the \(\frac{dy}{dx}\) symbol with its original algebraic formula to ensure the second derivative is expressed in terms of only \(x\) and \(y\).

Q: How do you use the Second Derivative Test to identify a local maximum in an implicit relation?

If \(\frac{dy}{dx} = 0\) and \(\frac{d^2y}{dx^2} < 0\) at a point \((a, b)\), the point is a local maximum.

Q: How do you use the Second Derivative Test to identify a local minimum in an implicit relation?

If \(\frac{dy}{dx} = 0\) and \(\frac{d^2y}{dx^2} > 0\) at a point \((a, b)\), the point is a local minimum.

AP Calculus BC · AP (Advanced Placement) · 4 min read · Updated: 7 Sept 2026

Introduction to Behaviors of Implicit Relations

Welcome! So far in Unit 5, you’ve learned how to analyze functions that look like \(y = f(x)\). But what happens when the relationship between \(x\) and \(y\) is a bit more "tangled"? Think of a circle like \(x^2 + y^2 = 25\) or a loopy curve like \(y^2 = x^3 + 3x\). These are implicit relations. Even though they might not pass the vertical line test (meaning they aren't technically functions), we can still use calculus to figure out exactly how they behave—where they go up, where they go down, and where they curve!

Did you know? Implicit relations are everywhere in the real world, from the orbits of planets to the way light reflects off a coffee mug (a shape called a caustic curve).

Note: If you need a refresher on the basics of finding local extrema or concavity for standard functions, check out Topics 5.4 and 5.6.

1. Finding Horizontal and Vertical Tangents

One of the most important behaviors to identify in an implicit relation is where the curve "levels out" or "drops off straight." We find these by looking at the derivative, \(\frac{dy}{dx}\).

Horizontal Tangents

A curve has a horizontal tangent line when the slope is zero. For a derivative written as a fraction, this happens when the numerator is zero (and the denominator is not zero).

Condition: Numerator of \(\frac{dy}{dx} = 0\)
What it looks like: The "top" or "bottom" of a loop or curve.

Vertical Tangents

A curve has a vertical tangent line when the slope is undefined or "infinite." This happens when the denominator is zero (and the numerator is not zero).

Condition: Denominator of \(\frac{dy}{dx} = 0\)
What it looks like: The "leftmost" or "rightmost" edges of a curve.

Common Mistake: If both the numerator and denominator are zero at the same point (the \(\frac{0}{0}\) case), the behavior is indeterminate! You might have a sharp point (cusp) or a place where the curve crosses itself. Always check both!

2. Determining Increasing and Decreasing Behavior

Just like with normal functions, the sign of the first derivative tells us the direction of the curve at a specific point \((x, y)\).

If \(\frac{dy}{dx} > 0\) at a point, the relation is increasing (moving up as you move right).
If \(\frac{dy}{dx} < 0\) at a point, the relation is decreasing (moving down as you move right).

Don't forget: Since these are implicit relations, your derivative will usually contain both \(x\) and \(y\). To find the slope, you must plug in both coordinates of the point.

3. Analyzing Concavity: The Second Derivative

To find out if an implicit curve is "smiling" (concave up) or "frowning" (concave down), we need the second derivative, \(\frac{d^2y}{dx^2}\). This is often the trickiest part for students because it requires the Chain Rule and Quotient Rule combined.

Step-by-Step: Finding the Second Derivative

Find the first derivative, \(\frac{dy}{dx}\), using implicit differentiation.
Differentiate \(\frac{dy}{dx}\) with respect to \(x\).
Crucial Step: Whenever you differentiate a term containing \(y\), you must multiply by \(\frac{dy}{dx}\).
Substitution Step: Replace the symbol \(\frac{dy}{dx}\) in your second derivative equation with the actual formula you found in Step 1.
Simplify to get the final expression for \(\frac{d^2y}{dx^2}\).

Quick Summary:
If \(\frac{d^2y}{dx^2} > 0\), the curve is concave up.
If \(\frac{d^2y}{dx^2} < 0\), the curve is concave down.

4. Using the Second Derivative Test for Local Extrema

We can use the Second Derivative Test to justify if a point with a horizontal tangent is a local maximum or minimum. This is a very common task on the AP Exam!

Suppose you found a point \((a, b)\) where \(\frac{dy}{dx} = 0\):

If \(\frac{d^2y}{dx^2} > 0\) at \((a, b)\), the curve is concave up. Therefore, \((a, b)\) is a local minimum.
If \(\frac{d^2y}{dx^2} < 0\) at \((a, b)\), the curve is concave down. Therefore, \((a, b)\) is a local maximum.

Analogy: Imagine a bowl. At the very bottom, the floor is flat (slope = 0). Because the bowl is curved upward (concave up), that flat spot must be a minimum!

5. Summary Checklist for Implicit Analysis

When you are given an implicit equation and asked to analyze its behavior at a point \((x, y)\):

Find the first derivative: Use implicit differentiation.
Find the slope: Plug the coordinates into \(\frac{dy}{dx}\).
Check for special tangents: Is the numerator zero (horizontal) or the denominator zero (vertical)?
Find the second derivative: Differentiate again, substituting the first derivative back in.
Identify concavity: Check the sign of \(\frac{d^2y}{dx^2}\).
Justify extrema: Use the signs of the first and second derivatives to conclude if a point is a max or min.

Key Takeaway: Analyzing implicit relations follows the exact same logic as analyzing regular functions. The only "extra" steps are using implicit differentiation and making sure to plug in both \(x\) and \(y\) values!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is an implicit relation in the context of calculus?

An implicit relation is a relationship between variables, such as \(x^2 + y^2 = 25\), where \(y\) is not isolated as a function of \(x\) and may not pass the vertical line test.

Under what condition does an implicit relation have a horizontal tangent line?

A horizontal tangent occurs when the numerator of the derivative \(\frac{dy}{dx}\) is zero, provided the denominator is not zero.

Under what condition does an implicit relation have a vertical tangent line?

A vertical tangent occurs when the denominator of the derivative \(\frac{dy}{dx}\) is zero, provided the numerator is not zero.

What does it mean if both the numerator and denominator of \(\frac{dy}{dx}\) are zero at a specific point?

If both are zero (the \(\frac{0}{0}\) case), the behavior is indeterminate, often indicating a sharp point (cusp) or where the curve crosses itself.

How do you determine if an implicit relation is increasing or decreasing at a specific point \((x, y)\)?

Plug both the \(x\) and \(y\) coordinates into \(\frac{dy}{dx}\); if the result is positive, the relation is increasing; if negative, it is decreasing.

What is the "Crucial Step" when finding the second derivative of an implicit relation?

Whenever you differentiate a term containing \(y\) with respect to \(x\), you must multiply by \(\frac{dy}{dx}\) due to the Chain Rule.

Why must you perform a "Substitution Step" when finding the second derivative expression?

You must replace the \(\frac{dy}{dx}\) symbol with its original algebraic formula to ensure the second derivative is expressed in terms of only \(x\) and \(y\).

How is concavity determined for an implicit relation?

If \(\frac{d^2y}{dx^2} > 0\), the curve is concave up; if \(\frac{d^2y}{dx^2} < 0\), the curve is concave down.

How do you use the Second Derivative Test to identify a local maximum in an implicit relation?

If \(\frac{dy}{dx} = 0\) and \(\frac{d^2y}{dx^2} < 0\) at a point \((a, b)\), the point is a local maximum.

How do you use the Second Derivative Test to identify a local minimum in an implicit relation?

If \(\frac{dy}{dx} = 0\) and \(\frac{d^2y}{dx^2} > 0\) at a point \((a, b)\), the point is a local minimum.

Ready to test yourself?

Turn these notes into exam-style practice. Get unlimited AI questions on this topic with instant marking and explanations.

Practise This Topic

More AP Calculus BC chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practise with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practising Now View Pricing

Done reading? Test yourself with AI practice questions

Practise This Topic Now