Related Application: Slopes of Implicitly Defined Curves

What This Application Is About

Not every curve in calculus is given as an explicit function such as . Sometimes the relationship between and is given by a single equation involving both variables.

For example, the circle

is an implicitly defined curve. Solving for gives two separate functions, but implicit differentiation lets us work with the original equation directly.

One of the most important applications is finding the slope of the curve at a particular point. The process is straightforward:

Differentiate both sides of the equation with respect to .
Collect all terms involving on one side.
Solve for .
Substitute the coordinates of the given point.

The Key Idea

When differentiating implicitly, remember that is treated as a function of . Therefore, whenever you differentiate a term involving , the chain rule appears.

That extra factor of is the key to implicit differentiation. For example:

The derivative of is therefore , not merely .

Finding the Slope at a Point

Suppose an implicitly defined curve is given by , and we want the slope at the point .

The derivative gives the slope of the tangent line at every point where the derivative exists. After finding the derivative formula, substitute the coordinates of the specified point.

Worked Examples

Example 1: Slope of a Circle

Find the slope of the circle at the point .

Step 1: Differentiate both sides with respect to :

Step 2: Differentiate each term:

Step 3: Solve for :

Step 4: Substitute :

Result: The slope at the point is .

Example 2: A More Complicated Polynomial Curve

Find the slope of the curve at the point .

Step 1: Differentiate both sides:

Step 2: Apply the power rule and chain rule:

Step 3: Collect the terms:

Step 4: Solve for :

Step 5: Substitute :

Result:

Example 3: Product Terms

Find the slope of the curve at the point .

Step 1: Differentiate both sides using the product rule on :

Step 2: Differentiate the remaining terms:

Step 3: Collect all terms involving :

Step 4: Solve for :

Step 5: Evaluate at the given point:

Result:

Example 4: Trigonometric Implicit Curve

Find the slope of the curve at the point .

Step 1: Differentiate both sides:

Step 2: Use the chain rule when differentiating :

Step 3: Solve for :

Step 4: Evaluate at the point:

Result:

From Slope to the Tangent Line

Once you know the slope at a particular point, you can immediately write the equation of the tangent line using point-slope form.

For example, if an implicitly defined curve passes through and the derivative gives , then the tangent line is:

This is a common AP Calculus application: implicit differentiation gives the slope, and the slope gives the tangent line.

How to Recognize the Problem on the AP Exam

Look for wording such as:

"Find the slope of the curve at the point..."
"Find the slope of the tangent line..."
"Find the equation of the tangent line..."
"At the point , find..."
"The curve is defined by the equation..." when both and appear in the same equation.

If the equation contains both variables and they cannot conveniently be separated, implicit differentiation is usually the intended method.

Common Mistakes

Forgetting the chain rule on y terms

The most important rule is that . Therefore , not simply . Missing the factor of produces the wrong slope.

Substituting the point too early

Usually, differentiate first and solve for before substituting the coordinates. Keeping the variables symbolic makes it much easier to see which terms belong on each side.

Forgetting that the requested point must lie on the curve

Before using a point, check that it actually satisfies the original equation. A point that is not on the curve cannot be used to find the slope of that curve.

Stopping after finding dy/dx

If the question asks for the slope at a specific point, the expression for is not yet the final answer. You still need to substitute the coordinates of the point.

Practice Problems

Question 1

Find the slope of the curve at the point .

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Question 2

Find the slope of the curve at the point .

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Question 3

Find the slope of the curve at the point .

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Question 4 — Tangent Line

Find the equation of the tangent line to at .

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