Second Derivatives of Parametric Equations

Differentiate the tangent slope and divide by the horizontal rate again to analyze local concavity.

College Board Topic 9.2 · CED p. 167 (PDF p. 172) · CED effective Fall 2020; project course outline 2026–27.

The second derivative with respect to x measures how the tangent slope changes as x changes. Start with $s (t) = d y / d x$ . Differentiate s with respect to t, then divide by x′ again: $\frac{d^{2} y}{d x^{2}} = \frac{d s / d t}{d x / d t}$ . The extra division is the chain rule. Merely differentiating s(t) gives ds/dt, which answers a different question.

Use this formula on intervals where x′ is nonzero and the needed derivatives exist. A positive second derivative means the graph, locally viewed as y versus x, is concave up; a negative one means concave down. That meaning is about horizontal position, not whether t is increasing or whether the point is moving up.

For $x = t^{2}$ , $y = t^{3}$ , with $t \neq 0$ , the slope is $3 t / 2$ and the second derivative is $3 / (4 t)$ . The branch with negative t is concave down, while the positive branch is concave up. At t=0 both coordinate rates vanish; our formula is not a license to assign a finite second derivative there.

Let's work through it

With $x = t^{2}$ , $y = t^{3}$ , compute $d y / d x = 3 t / 2$ for $t \neq 0$ . Why: Divide y′=3t² by x′=2t only where the denominator is nonzero.
Differentiate the slope to get $d s / d t = 3 / 2$ . Why: This is the change in slope per unit parameter.
Divide again: $d^{2} y / d x^{2} = 3 / (4 t)$ , so at $t = 2$ it is $3 / 8$ . Why: A change per unit x requires division by dx/dt=2t.

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Question 1 of 20 self-assessed

Question 1

d s / d t = 6

and

x' = 3

, what is

d^{2} y / d x^{2}

Your answer and reasoning

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Question 1: prompt and worked reasoning

d s / d t = 6

and

x' = 3

, what is

d^{2} y / d x^{2}

2

Divide the slope rate by the horizontal rate.

Question 2: prompt and worked reasoning

For the example above, what is the concavity on

t < 0

Concave down.

3/(4t) is negative on that interval.

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Retrieval 1

d s / d t = 6

and

x' = 3

, what is

d^{2} y / d x^{2}

Worked answer and reason

2

Divide the slope rate by the horizontal rate.

Retrieval 2

For the example above, what is the concavity on

t < 0

Worked answer and reason

Concave down.

3/(4t) is negative on that interval.

Catch the tempting mistake

The second derivative is d/dt of dy/dx.: That is only the numerator. Divide by dx/dt to change from a t-rate to an x-rate.

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Unit 9 problem 4 · No calculator · analytic
Unit 9 problem 5 · No calculator · analytic
Unit 9 problem 6 · No calculator · verbal
Unit 9 problem 20 · No calculator · analytic

For the teacher

Have students label ds/dt and ds/dx separately before simplifying. Include a backwards-moving branch so concavity is not confused with orientation.

Unit planning and formative guidance →

Glossary and deeper reading · 4 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Second derivative
A second derivative is the derivative of the first derivative. It measures how the first rate of change changes with the input; further repeated differentiation gives higher-order derivatives. For example, f(x)=x³ has f′(x)=3x² and f″(x)=6x. For position measured against time, the second derivative is acceleration.
Read more: 3.6 · Second derivative →
Chain rule
The chain rule differentiates a composition by multiplying the outer derivative at the inner output by the inner derivative. For differentiable functions, (f(g(x)))′=f′(g(x))g′(x). For example, the derivative of sin(x²) is cos(x²)·2x. Nested compositions require a factor for each successive layer.
Read more: 3.1 · Chain rule →
Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
Read more: 2.2 · Derivative →
Concavity
Concavity describes how a graph’s slope changes as the input increases. A graph is concave up where its derivative increases and concave down where its derivative decreases. For example, x² is concave up because its derivative 2x increases. A positive second derivative establishes concave-up behavior on an interval, while a negative one establishes concave-down behavior.
Read more: 5.6 · Concavity →

Let's work through it

Try it from memory

Try it before you peek.

Question 1

Retrieval 1

Retrieval 2

Catch the tempting mistake

Put it to work

Second derivative

Chain rule

Derivative

Concavity

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