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Higher Derivative

Higher Derivative

Any derivative beyond the first derivative. That is, the second, third, fourth, fifth etc. derivatives.

 

 

See also

nth derivative

Key Formula

f(n)(x)=dndxnf(x)f^{(n)}(x) = \frac{d^n}{dx^n}f(x)
Where:
  • f(x)f(x) = The original function
  • nn = The order of the derivative (n = 2 for the second derivative, n = 3 for the third, etc.)
  • f(n)(x)f^{(n)}(x) = The nth derivative of f with respect to x

Worked Example

Problem: Find the first four derivatives of f(x) = x⁵ − 3x³ + 2x.
Step 1: Find the first derivative by applying the power rule to each term.
f′(x)=5x4−9x2+2f'(x) = 5x^4 - 9x^2 + 2
Step 2: Differentiate again to get the second derivative.
f′′(x)=20x3−18xf''(x) = 20x^3 - 18x
Step 3: Differentiate once more for the third derivative.
f′′′(x)=60x2−18f'''(x) = 60x^2 - 18
Step 4: Differentiate one final time for the fourth derivative.
f(4)(x)=120xf^{(4)}(x) = 120x
Answer: The four higher-order derivatives are f'(x) = 5x⁴ − 9x² + 2, f''(x) = 20x³ − 18x, f'''(x) = 60x² − 18, and f⁽⁴⁾(x) = 120x. Notice that each derivative reduces the degree of the polynomial by one.

Another Example

Problem: Find the first three derivatives of g(x) = sin(x).
Step 1: Differentiate sin(x) to get the first derivative.
g′(x)=cos⁡(x)g'(x) = \cos(x)
Step 2: Differentiate cos(x) to get the second derivative.
g′′(x)=−sin⁡(x)g''(x) = -\sin(x)
Step 3: Differentiate −sin(x) to get the third derivative.
g′′′(x)=−cos⁡(x)g'''(x) = -\cos(x)
Answer: The derivatives of sin(x) cycle through a repeating pattern: cos(x), −sin(x), −cos(x), sin(x), and then the cycle repeats. This means the fourth derivative brings you back to sin(x).

Frequently Asked Questions

Do higher derivatives always exist?
Not necessarily. A function might be differentiable once but not twice. For example, f(x) = x|x| has a first derivative everywhere, but its second derivative does not exist at x = 0. A function that can be differentiated infinitely many times is called "infinitely differentiable" or "smooth."
What does the third derivative mean physically?
If the function represents position over time, the first derivative is velocity, the second is acceleration, and the third derivative is called "jerk" — it measures how quickly acceleration is changing. Engineers care about jerk because sudden changes in acceleration cause discomfort or mechanical stress.

First Derivative vs. Higher Derivative

The first derivative f'(x) measures the instantaneous rate of change of the original function. Higher derivatives measure rates of change of rates of change: f''(x) tells you how f'(x) is changing, f'''(x) tells you how f''(x) is changing, and so on. The first derivative gives slope; the second derivative gives concavity; further derivatives reveal increasingly subtle aspects of a function's behavior.

Why It Matters

Higher derivatives are essential in physics, where position, velocity, acceleration, and jerk form a chain of successive derivatives. In mathematics, Taylor series use all higher derivatives of a function at a single point to reconstruct the entire function as an infinite polynomial. Engineers also rely on higher derivatives to analyze stability, curvature, and the smoothness of curves in design.

Common Mistakes

Mistake: Confusing notation: writing f²(x) instead of f''(x), which would mean f(f(x)).
Correction: Use prime marks f'', f''' for low orders, and parenthesized superscripts f⁽⁴⁾, f⁽⁵⁾ for fourth order and above. The notation f⁽ⁿ⁾(x) always means the nth derivative, not f raised to a power.
Mistake: Forgetting to apply the chain rule or product rule at each stage when differentiating composite or product expressions multiple times.
Correction: Each differentiation step is a full derivative on its own. Apply all relevant differentiation rules at every step — do not assume the structure simplifies just because you already differentiated once.

Related Terms

  • Derivative — The foundational concept that higher derivatives extend
  • First Derivative — The starting point before higher derivatives
  • Second Derivative — The most commonly used higher derivative
  • nth Derivative — General notation for any order derivative
  • Power Rule — Key rule applied repeatedly for polynomial derivatives
  • Concavity — Determined by the sign of the second derivative
  • Taylor Series — Uses all higher derivatives to approximate functions