Key takeaways
- A higher-order derivative means differentiating again and again.
- The k-th derivative of xⁿ is n!/(n − k)! × xⁿ⁻ᵏ.
- The second derivative is not the first derivative squared.
- Position → speed → acceleration: first and second derivatives.
- For xⁿ, every derivative past the n-th is 0.
In this guide
What is a higher-order derivative?
The first derivative, f′(x), is the slope or rate of change. Differentiate f′(x) and you get the second derivative, f″(x). Do it again for the third derivative, f‴(x), and so on. The k-th derivative is written dᵏy/dxᵏ.
If f is the position of a car, f′ is its speed and f″ is its acceleration (how fast the speed is changing).
Why do higher-order derivatives matter?
The first derivative tells you whether something is rising. The second tells you whether it is speeding up or slowing down. That difference matters: a company whose sales are rising but slowing needs a different plan from one whose sales are rising faster and faster.
Lift and train designers limit “jerk”, the third derivative of position, because sudden changes in acceleration are what make passengers feel uncomfortable.
The formula
For powers of x, applying the power rule k times gives:
| Symbol | What it means | Example |
|---|---|---|
| n | The power of x (a whole number) | 3 |
| k | How many times you differentiate (0 ≤ k ≤ n) | 2 |
| n!/(n − k)! | The numbers you bring down, multiplied: n × (n − 1) × … | 3 × 2 = 6 |
| x | The point where you evaluate the result | 2 |
The ! sign is a factorial: 4! = 4 × 3 × 2 × 1 = 24. See the factorial calculator.
How to find a higher-order derivative step by step
- 1Take the first derivative. Use the power rule: bring the power down, subtract 1.
- 2Differentiate the answer again. Repeat until you have done it k times.
- 3Keep multiplying the numbers. The coefficient builds up: 5, then 5 × 4, then 5 × 4 × 3.
- 4Put in the value of x. Work out the final number.
Now try it yourself. Enter the power n, the order k and a value of x in the calculator below.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting to multiply all the brought-down numbers. Each step multiplies the coefficient again.
- Asking for k bigger than n. Once you pass the n-th derivative of xⁿ you only get 0. The calculator asks for whole numbers with k ≤ n.
- Mixing up the notation. f″(x) and d²y/dx² mean the same thing. d²y/dx² is not (dy/dx)².
Tips and tricks
- Polynomials: differentiate each term separately. A cubic has a constant third derivative and a zero fourth derivative.
- Max or min? The second derivative tells you whether a critical point is a maximum or a minimum.
- Series: Taylor series are built from every derivative of a function at one point.
Where you’ll use it in real life
- Physics: acceleration is the second derivative of position (see the acceleration calculator).
- Economics: the second derivative shows whether growth is speeding up or slowing down.
- Engineering: limiting jerk (the third derivative) in lifts, trains and roller coasters.
- Graph sketching: concave up or down, and where the bend changes direction.
Quick summary
A higher-order derivative is what you get by differentiating more than once. For xⁿ, the k-th derivative is n!/(n − k)! × xⁿ⁻ᵏ. Use the calculator above to find any order at any point, with every step shown.
Worked example
A ball rolls down a ramp and its distance is s = t³ metres after t seconds. The second derivative of distance is acceleration. What is the acceleration (in m/s²) at t = 2 seconds? Here n = 3, k = 2 and x = 2.
- 01Start: f(x) = x^3
- 02Apply power rule 2 times:
- 03d^2/dx^2(x^3) = 6x^1
- 04Coefficient = 3×2×…×2 = 6
- 05At x = 2: 6 × 2^1 = 6 × 2
- 06= 12
These numbers are pre-filled in the calculator above.
Frequently asked questions
What does the second derivative tell you?
It tells you how the slope is changing. Positive means the curve bends upward (concave up), negative means it bends downward (concave down).
What is the second derivative of x²?
The first derivative is 2x and the second is 2. A constant second derivative means the curve bends by the same amount everywhere.
How do I write higher-order derivatives?
Use f″(x) or d²y/dx² for the second, f‴(x) or d³y/dx³ for the third, and f⁽ⁿ⁾(x) or dⁿy/dxⁿ for the n-th.
