Second Derivatives of Parametric Equations

Why Second Derivatives Are Different for Parametric Equations

With an ordinary function , finding the second derivative means differentiating with respect to one more time. Parametric equations work differently because and are both given in terms of a third variable, usually called .

Suppose we have:

We already know that the first derivative is found using:

But to find the second derivative, we cannot simply differentiate the numerator and denominator separately. Instead, we must differentiate the first derivative with respect to , and then divide by .

The Second-Derivative Formula

The fundamental formula to memorize is:

Another way to write the same formula is:

The idea is simple: differentiate with respect to , then divide by .

Why Do We Divide by ?

This is the step that often feels strange at first. Remember that the first derivative is:

So is itself a function of . To differentiate it with respect to , we use the chain rule:

Since , we have:

Solving this relationship for the second derivative gives the standard parametric formula.

A Step-by-Step Method

Find . Use the parametric first-derivative formula.
Differentiate with respect to . Treat the entire first derivative as a function of .
Divide by . This converts the derivative with respect to into the derivative with respect to .
Evaluate at the requested parameter value, if the problem gives one.

Worked Examples

Example 1: Basic Polynomial Parametric Equations

Suppose:

Find the second derivative at .

Step 1: Find the first derivative:

Step 2: Differentiate the first derivative with respect to :

Step 3: Find :

Step 4: Apply the second-derivative formula:

Step 5: Evaluate at :

Result:

Example 2: A Rational First Derivative

Let:

Find at .

Step 1: Find the first derivative:

Step 2: Differentiate with respect to . Use the quotient rule:

Step 3: Find :

Step 4: Divide by :

Step 5: Evaluate at :

Result:

Example 3: Finding Concavity

Suppose:

Determine whether the curve is concave up or concave down at .

Step 1: Find the first derivative:

Step 2: Differentiate with respect to :

Step 3: Find :

Step 4: Calculate the second derivative:

Step 5: Evaluate at :

Conclusion:

Example 4: AP-Style Reasoning

A curve is defined parametrically by:

At , determine whether the curve is concave up or concave down.

Step 1: Find the first derivative:

Step 2: Differentiate with respect to :

Step 3: Find :

Step 4: Apply the formula:

Step 5: Evaluate:

Conclusion:

Second Derivative and Concavity

One of the most important reasons the AP exam asks for second derivatives of parametric equations is to determine concavity.

The interpretation is exactly the same as for ordinary functions:

So once you calculate , you can determine the shape of the curve at a particular parameter value.

If , the curve is concave up.
If , the curve is concave down.
If , the test by itself does not determine concavity.

The Most Important AP Calculus Shortcut

On an AP problem, you will often see something like:

You do not need to eliminate the parameter and rewrite the equation as . In fact, doing so is usually unnecessary and can make the problem much harder.

Stay in parametric form. Find:

and then divide by .

A Reliable Memory Trick

Think of the process as:

First derivative: divide by .
Second derivative: differentiate that result with respect to , then divide by again.

Common Mistakes

Forgetting the final division by

This is the most common error. Finding the derivative of with respect to only gives . You still have to divide by to obtain .

Differentiating with respect to incorrectly

Once you have , remember that you are differentiating it with respect to , not directly with respect to . Treat as the variable throughout this step.

Using instead of

The denominator in the second-derivative formula is , the derivative of with respect to . Do not accidentally use itself.

Trying to eliminate the parameter

You almost never need to solve for in terms of . The parametric derivative formulas are specifically designed so that you can work directly with and .

Practice Problems

Question 1

Given , find at .

Show answer

Question 2

Given , find at .

Show answer

Question 3

A curve is defined by . Determine whether the curve is concave up or concave down at .

Show answer

Question 4 (AP-Style)

For , determine the value of at .

Show answer

Test yourself: AP Calculus practice exams

Ready to check what you know? Work an AP-style practice exam, then open the full solutions.

AB Multiple Choice AB Free Response BC Multiple Choice BC Free Response

Want all 1,114 practice problems?

This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.