HesapHanem

Higher Order Derivative Calculator

Enter the polynomial coefficients and the derivative order to compute the higher-order derivative.

Result

How it is calculated

The polynomial is entered as comma-separated coefficients, highest power first, with a 0 for every missing power: 3,-2,5,1,7 means 3x⁴ − 2x³ + 5x² + x + 7. Each differentiation uses the power rule d/dx (c·xk) = c·k·xk−1: every coefficient is multiplied by its power and the powers drop by one, and the constant term disappears. The tool repeats this n times, where n is the order (a whole number, 0 or more; 0 returns the polynomial unchanged). Once the order exceeds the degree the answer is 0. The output is written with superscript exponents and a degree. Only plain numbers separated by commas are read: spaces alone, x^2 notation or a decimal order such as 1.5 return no result.

Worked example

Differentiate 3x⁴ − 2x³ + 5x² + x + 7 twice. First derivative: 12x³ − 6x² + 10x + 1 (3×4, −2×3, 5×2, 1×1, the constant 7 drops). Second derivative: 36x² − 12x + 10 (12×3, −6×2, 10×1, the constant 1 drops). The tool shows 36x² - 12x + 10, degree 2, order 2. A fifth derivative of this degree-4 polynomial would be 0.

Try this example →

Common mistakes

  • Skipping the zero for a missing power. For 2x³ − 4 you must type 2,0,0,-4; typing 2,-4 means 2x − 4 and the derivative is just 2.
  • Entering the coefficients lowest power first (7,1,5,-2,3); the tool reads the first number as the leading coefficient.
  • Typing the polynomial as text such as 3x^4-2x^3+7, or using spaces instead of commas; the tool needs just numbers separated by commas.
  • Confusing the order with the power: order 2 means differentiate twice, not raise to the power 2.
  • Expecting the value of the derivative at a point. The tool returns the derivative polynomial; substitute x yourself afterwards.

About this calculator

The nth derivative of a polynomial tells you how a rate of change is itself changing — the 2nd derivative gives acceleration when the 1st gives velocity, and so on. This calculator applies the power rule — d/dx[x^n] = n·x^(n−1) — repeatedly, n times, to a polynomial entered as a list of coefficients from the highest degree down.

Each application drops the degree by one and multiplies the coefficient by the current exponent, so a degree-5 polynomial differentiated 6 times collapses to zero — a quick way to see that the derivative order can't exceed the polynomial's degree without vanishing. This mirrors what a computer algebra system does symbolically, just done term by term.

It's useful for calculus students checking homework on higher-order derivatives, Taylor series coefficients, or curve concavity, and for anyone verifying a symbolic differentiation by hand.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    Higher derivatives have physical meaning: if the coefficients describe position over time, the first derivative is velocity, the second acceleration and the third jerk. A useful check is that the n-th derivative of a degree n polynomial is n factorial times the leading coefficient, so 2x³ - 3x + 5 gives 12 at order 3. Post a polynomial whose answer surprised you.

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators