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Higher-Order Derivatives Calculator

Higher-derivative calculator. Enter f(x), order n and x to compute f^(n)(x) by repeated central differences—for Taylor preparation.

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f^(n)(x) = the nth derivative of f
Supports + - * / ^ and sin cos tan sqrt exp log pi e
Integer 1~5
Compute the n-th derivative here

📖 Tutorial｜Higher-Order Derivatives

1. Definition
Higher-order derivatives are repeated differentiation: f″=(f′)′, f‴=(f″)′, denoted f^(n)(x).
2. Symbols
NotationMeaning
f′First derivative
f″Second derivative
f‴Third derivative
f^(n)n-th derivative
3. How it works
  • We recursively apply central differences;
  • Each extra order differentiates the result again;
  • The value f^(n)(x) is output.
4. Steps
  1. Enter f(x);
  2. Enter order n and x;
  3. Click Calculate.
5. Example
Example: Every derivative of eˣ is eˣ, so f^(3)(1)=e≈2.718.
6. Pitfalls
Numerical error accumulates at high order;
Polynomial derivatives eventually become constants;
The second derivative decides concavity and extrema.

❓ FAQ｜Higher-Order Derivatives

What is the second derivative for?
Concavity (f″>0 up, f″<0 down) and the second-derivative test for extrema.
What are the higher derivatives of eˣ?
Still eˣ—a special property of the exponential.
Why the error at high order?
Each numerical difference adds error; n orders accumulate n times.
Fourth derivative of sin x?
It cycles: sin→cos→−sin→−cos→sin, back to sin x.
Sixth derivative of x^5?
The fifth is 120; the sixth and above are 0.