Behavior of Implicit Relations Using Derivatives

Why Implicit Relations Need a Different Approach

In many calculus problems, a function is written explicitly as . That makes it easy to calculate and use its sign to determine whether the function is increasing or decreasing.

But not every curve is given that way. A circle, for example, is naturally written as . Solving for gives two separate functions, so treating the equation as a single explicit function can be awkward.

Implicit differentiation lets us work directly with the relation. Once we find , we can analyze the behavior of the curve just as we would analyze an ordinary function.

The key idea is simple: the sign of tells us the direction in which the curve is moving as increases.

What "Increasing" and "Decreasing" Mean for an Implicit Curve

For an explicitly defined function , we say the function is increasing on an interval when throughout that interval. Likewise, it is decreasing when .

The same idea applies locally to an implicitly defined relation. At a point where the curve has a well-defined vertical change with respect to horizontal change:

So the sign chart is interpreted exactly as you already know:

• : the curve rises as you move from left to right.
• : the curve falls as you move from left to right.
• : the curve has a horizontal tangent, which may indicate a local maximum, local minimum, or another horizontal-tangent point.

Step 1: Find the Implicit Derivative

The first step is always to differentiate the entire relation with respect to . Remember that is a function of , even if the equation doesn't explicitly write .

For example, consider:

Differentiate both sides with respect to . The derivative of is , while the derivative of requires the chain rule.

Now solve algebraically for :

Once this expression is available, we can determine where the curve is increasing or decreasing by analyzing its sign.

Step 2: Find Critical Points

For an implicitly defined relation, horizontal tangent points occur when , provided the derivative exists there.

For a derivative written as a fraction, such as , there are two especially important possibilities:

1. The numerator is zero, giving .
2. The denominator is zero, making undefined and potentially indicating a vertical tangent.

These are not automatically maxima or minima. They are candidate points that must be analyzed using the behavior of the derivative on either side.

Worked Example 1: Increasing and Decreasing on a Circle

Step 3: Use a Sign Chart

A sign chart is one of the most reliable ways to analyze the behavior of an implicitly defined curve.

Suppose your derivative has the form:

Find the values of where the numerator or denominator is zero. These values divide the domain into intervals. Then test the sign of in each interval.

Worked Example 2: Finding a Local Maximum and Minimum

Horizontal Tangents vs. Vertical Tangents

Implicit differentiation is especially useful because it can reveal tangent behavior that is difficult to see from an explicit equation.

If:

then a horizontal tangent is possible because the slope is zero.

If:

then the slope is undefined. If the numerator is nonzero and the denominator approaches zero, this can correspond to a vertical tangent.

Worked Example 3: Identifying a Vertical Tangent

How Derivatives Describe the Shape of an Implicit Curve

The derivative contains much more information than just the slope at one point. By studying its sign, you can describe how an implicit curve behaves across an interval.

• Positive derivative: the curve moves upward as you move right.
• Negative derivative: the curve moves downward as you move right.
• Derivative equal to zero: horizontal tangent.
• Derivative undefined: possible vertical tangent or another special point.
• Sign change from positive to negative: local maximum.
• Sign change from negative to positive: local minimum.

This means that the familiar first-derivative analysis from explicit functions still works. The only additional step is that you must first obtain using implicit differentiation.

Second Derivatives and Concavity

You can also use implicit differentiation to study concavity. Once you have an expression for , differentiate it again with respect to .

The sign of can then be used to determine concavity, just as it is for an explicitly defined function:

• : the curve is concave upward.
• : the curve is concave downward.

Be careful: for implicit relations, the algebra involved in finding can be substantially more complicated because itself appears inside the expression.

AP Calculus Problem-Solving Workflow

When an AP Calculus question asks you to analyze the behavior of an implicitly defined relation, use this workflow:

1. Differentiate implicitly. Treat as a function of .
2. Solve for if the problem asks for a derivative or requires sign analysis.
3. Find candidate critical values. Look for where is zero or undefined.
4. Use the original relation to determine the corresponding points on the curve.
5. Build a sign chart for .
6. Classify behavior. Identify increasing/decreasing intervals, local extrema, and possible vertical tangents.
7. State your answer in terms of the requested variable and interval.

Common Mistakes

Practice Problems