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nth Derivative — Definition, Formula & Examples

nth Derivative

The result of taking the derivative of the derivative of the derivative etc. of a function a total of n times. Written
f (n)(x) or .

Note: f (0)(x) is the same thing as f(x).

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 being differentiated
  • nn = A non-negative integer indicating how many times to differentiate
  • f(n)(x)f^{(n)}(x) = The function after being differentiated n times

Worked Example

Problem: Find the 4th derivative of f(x) = x⁵.
Step 1: Take the first derivative.
f′(x)=5x4f'(x) = 5x^4
Step 2: Take the second derivative.
f′′(x)=20x3f''(x) = 20x^3
Step 3: Take the third derivative.
f′′′(x)=60x2f'''(x) = 60x^2
Step 4: Take the fourth derivative.
f(4)(x)=120xf^{(4)}(x) = 120x
Answer: The 4th derivative is f⁽⁴⁾(x) = 120x.

Why It Matters

The second derivative reveals concavity and acceleration, and higher-order derivatives appear throughout physics (e.g., the third derivative of position is called "jerk"). Taylor series use nth derivatives to approximate functions as polynomials, making them essential in both applied and theoretical mathematics.

Common Mistakes

Mistake: Confusing the notation f⁽ⁿ⁾(x) with raising f(x) to the nth power.
Correction: The parentheses around n indicate repeated differentiation, not exponentiation. For example, f⁽³⁾(x) means the third derivative, not [f(x)]³.

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