Higher Derivatives
Textbook Reference
| Primary source | OpenStax Calculus Volume 1, Section 3.2: “The Derivative as a Function” |
| Book URL | https://openstax.org/details/books/calculus-volume-1 |
Freely available and openly licensed.
Try This First: Derivative of a Derivative
For $f(x) = x^4$, the first derivative is $f'(x) = 4x^3$.
Now $f'(x) = 4x^3$ is itself a function. What is its derivative?
Predict: $(4x^3)' = $ $\underline{\hspace{2cm}}$.
This is the second derivative of $f$. It measures how the slope itself is changing: is the graph of $f$ becoming steeper or more gentle as $x$ increases?
Quantity-First Framing
A derivative is a function, so you can differentiate it again. The second derivative $f''(x)$ tells you how the slope of the original curve is changing. In physics, if $s(t)$ is position, then:
- $v(t) = s'(t)$ is velocity (rate of change of position)
- $a(t) = s''(t)$ is acceleration (rate of change of velocity)
The derivative chain can continue indefinitely: $f'''$, $f^{(4)}$, and so on.
Prerequisite Check
Quick Reference
Notation:
| Order | Prime | Leibniz | Description |
|---|---|---|---|
| 1st | $f'(x)$ | $\dfrac{dy}{dx}$ | Rate of change of $f$ |
| 2nd | $f''(x)$ | $\dfrac{d^2y}{dx^2}$ | Rate of change of $f'$; concavity |
| 3rd | $f'''(x)$ | $\dfrac{d^3y}{dx^3}$ | Rate of change of acceleration (jerk) |
| $n$th | $f^{(n)}(x)$ | $\dfrac{d^ny}{dx^n}$ | Apply differentiation $n$ times |
Key Concepts
1. Computing Higher Derivatives
Example 1. Find $f'''(x)$ for $f(x) = x^5 - 3x^3 + 2x$.
$f'(x) = 5x^4 - 9x^2 + 2$.
$f''(x) = 20x^3 - 18x$.
$f'''(x) = 60x^2 - 18$.
For a degree-$n$ polynomial, the $n$th derivative is a constant; all higher derivatives are zero.
Example 2. Find all derivatives of $f(x) = \sin x$.
$f'(x) = \cos x$. $f''(x) = -\sin x$. $f'''(x) = -\cos x$. $f^{(4)}(x) = \sin x$.
The pattern repeats with period 4. So $f^{(n)}(x)$ depends only on $n \bmod 4$.
2. Two Representations: Physics and Graph
Physics meaning. For $s(t) = t^3 - 6t^2 + 9t$ (position in meters, $t$ in seconds):
$v(t) = s'(t) = 3t^2 - 12t + 9$ (velocity, m/s).
$a(t) = s''(t) = 6t - 12$ (acceleration, m/s$^2$).
$a(2) = 0$: velocity has a local minimum/maximum at $t = 2$.
Graph meaning. The sign of $f''(x)$ reveals concavity:
| $f''(x)$ | Meaning for slopes | Geometric shape |
|---|---|---|
| $> 0$ | Slope is increasing | Concave up (cup-shaped) |
| $= 0$ | Slope momentarily flat | Possible inflection |
| $< 0$ | Slope is decreasing | Concave down (cap-shaped) |
Translation prompt. For $f(x) = x^2$: $f'(x) = 2x$ (slopes grow as $x$ increases) and $f''(x) = 2 > 0$ (concave up everywhere). Does this match the graph? Yes: the parabola $y = x^2$ is bowl-shaped.
3. The $n$th Derivative of $x^n$
For $f(x) = x^n$ (positive integer $n$):
$f^{(k)}(x) = n(n-1)\cdots(n-k+1)\,x^{n-k}$ for $k \leq n$.
$f^{(n)}(x) = n!$ (a constant).
$f^{(n+1)}(x) = 0$.
So $f^{(n)}(x^n) = n!$ -- the factorial of the degree. Every polynomial eventually differentiates to zero.
4. Ask Why: Why Does $f'' > 0$ Mean Concave Up?
If $f''(x) > 0$ on an interval, then $f'(x)$ is increasing there: slopes are growing as $x$ increases. The graph is curving upward -- bending toward higher slopes -- which is exactly “concave up.”
Concavity is not about whether $f$ is positive or increasing. It is purely about whether the slope is increasing ($f'' > 0$) or decreasing ($f'' < 0$). A decreasing function can be concave up, and an increasing function can be concave down.
Named Misconception: height-vs-slope
$f''(a) > 0$ does NOT mean $f(a) > 0$ or $f'(a) > 0$. It only means the slope is increasing at $a$.
One way to see the error breaks: take $f(x) = x^2 - 10$. At $x = 0$: $f(0) = -10 < 0$, $f'(0) = 0$, and $f''(0) = 2 > 0$. The function is negative and has zero slope -- but it is concave up.
Height, slope, and concavity are three independent facts, read from $f$, $f'$, and $f''$ separately.
Common Errors
| Error | Specific example | Correction |
|---|---|---|
| Writing second derivative as a square | “$f^2(x)$” for $f''(x)$ | $f^2(x) = [f(x)]^2$; second derivative is $f''$ or $f^{(2)}$ |
| Leibniz notation as a square | $\left(\frac{dy}{dx}\right)^2$ for $\frac{d^2y}{dx^2}$ | $\frac{d^2y}{dx^2}$ means differentiate twice, not square the first derivative |
| Stopping after one derivative | Computing $f'$ when $f''$ is requested | Apply the differentiation rule a second time to $f'$ |
Leveled Practice
Level 1 -- Computing Higher Derivatives
Problem 1. Find $f''(x)$ for $f(x) = 4x^3 - 6x^2 + 5$.
Show answer
$f'(x) = 12x^2 - 12x$. $f''(x) = 24x - 12$.
Problem 2. Find the fourth derivative of $g(x) = e^{2x}$.
Show answer
$g'(x) = 2e^{2x}$; $g''(x) = 4e^{2x}$; $g'''(x) = 8e^{2x}$; $g^{(4)}(x) = 16e^{2x}$.
Pattern: $g^{(n)}(x) = 2^n e^{2x}$.
Level 2 -- Physics and Concavity
Problem 3. For $s(t) = t^3 - 9t^2 + 24t$: find $v(t)$ and $a(t)$. When is acceleration zero?
Show answer
$v(t) = 3t^2 - 18t + 24$. $a(t) = 6t - 18$. $a(t) = 0$ at $t = 3$ s.
Problem 4. For $f(x) = x^3 - 3x$: find $f''$ and determine where the graph is concave up.
Show answer
$f'(x) = 3x^2 - 3$. $f''(x) = 6x$.
Concave up ($f'' > 0$) when $x > 0$. Concave down ($f'' < 0$) when $x < 0$.
Level 3 -- Low-Floor-High-Ceiling Extension
Problem 5 (Extension).
(a) (Floor) Compute $f^{(4)}(x)$ for $f(x) = \sin x$. What is $f^{(100)}(x)$?
(b) (Mid) For $f(x) = x^n$ (positive integer $n$), what is $f^{(n)}(x)$? And $f^{(n+1)}(x)$? What does this say about the degree of the $k$th derivative of a degree-$n$ polynomial?
(c) (Ceiling) Show that $y = e^x$ satisfies $y^{(n)} = y$ for all $n \geq 1$, and argue informally why this makes $e^x$ special compared to all other functions.
Show answer
(a) The $\sin x$ cycle has period 4: $\sin\to\cos\to-\sin\to-\cos\to\sin$. $f^{(4)} = \sin x$. Since $100 = 4 \times 25$: $f^{(100)}(x) = \sin x$.
(b) $f^{(n)}(x) = n!$; $f^{(n+1)} = 0$. The $k$th derivative of a degree-$n$ polynomial has degree $n - k$ (for $k \leq n$), and the $(n+1)$th derivative vanishes.
(c) $(e^x)' = e^x$, so $(e^x)'' = e^x$, and by induction $(e^x)^{(n)} = e^x$ for all $n$.
What makes $e^x$ special: it is its own derivative. The chain of derivatives cycles back to the original. Every other elementary function eventually changes under repeated differentiation ($x^n$ hits zero; $\sin x$ cycles through 4 functions). Only $e^x$ stays fixed at every step.
Common Misconceptions
$f''(x)$ tells how fast $f(x)$ is changing. The second derivative $f''(x)$ is the derivative of $f'(x)$, so it tells how fast the slope of $f$ is changing, not how fast $f$ itself is changing. That role belongs to $f'(x)$. In motion problems: $f'$ is velocity (how fast position changes) and $f''$ is acceleration (how fast velocity changes). Confusing these leads to errors in concavity analysis and motion problems.
if $f''(a) = 0$ then $a$ is an inflection point. An inflection point requires $f''$ to change sign near $a$, not merely equal zero there. The function $f(x) = x^4$ has $f''(0) = 0$, but $f''(x) = 12x^2 \ge 0$ everywhere, so the concavity never changes at $x = 0$ and there is no inflection point there. Zero second derivative is a necessary condition to check, not a sufficient conclusion.
Mastery Checklist
Mental Model
Each derivative strips away one “layer” of information: $f$ tells you the value; $f'$ tells you the rate; $f''$ tells you how the rate is changing. The chain can continue, but in practice, the first three layers (value, slope, concavity) answer most questions in Calculus I.
For polynomials, the chain terminates. For $e^x$, it cycles back. For trig functions, it cycles with period 4. This behavior at higher derivatives is the first hint of why Taylor series -- which encode a function by ALL its derivatives at a point -- are so powerful.
Connections
Within MATH161
- Concavity and the Second Derivative Test: $f''$ at a critical point determines whether that point is a local min, max, or saddle.
- Physics: Position $\to$ velocity $\to$ acceleration is the primary motivation for higher derivatives in applied science.
- Taylor polynomials (MATH162): The $n$th-order Taylor polynomial uses all derivatives up to order $n$ at a single point.
Back to Calculus I Skills | Previous: The Derivative Function | Next: Derivative Notations