Second Derivative Calculator

What the second derivative measures

The first derivative f′ gives slope or instantaneous rate of change. The second derivative f′′ tells how that slope itself changes — concavity of a graph, acceleration from a position function, or curvature clues in applied models. A second derivative calculator differentiates twice automatically and shows each rule along the way.

Searchers also look for a double derivative calculator or 2nd derivative calculator; they mean the same computation. Raise Order above 2 on this page when a problem asks for f⁽³⁾ or higher (supported through 5th order).

How to use this double derivative tool

1. Enter a function of x (or change the variable).
2. Confirm Order is 2 — already selected for this landing page.
3. Press Calculate to see the simplified second derivative and the step list.
4. Compare each labeled rule with your handwritten work for exam prep.

Need only the first derivative? Switch Order to 1 or use the main derivative calculator. Want every rule emphasized for teaching? Open the derivative calculator with steps.

Second derivative examples

Polynomials and powers

• x^3 — f′ = 3x², f′′ = 6x
• 1/x or x^(-1) — negative powers twice
• sqrt(x) — fractional power, then differentiate again

Trig, exp, and products

• sin(x) / cos(x) — classic second-derivative cycle
• e^x — second derivative equals the function
• x*ln(x) — product rule on the first pass, then simplify for f′′
• x^2*sin(x) — product + chain; steps help audit the second pass

Concavity, inflection, and applications

Sign of f′′ indicates concavity: positive often means concave up, negative concave down (on intervals where f′′ exists). Zeros of f′′ are candidates for inflection points after you check a sign change. In physics, if s(t) is position, s′′(t) is acceleration — compute the second derivative symbolically here, then plug in a time value on paper.

For a numerical slope at one x-value after differentiating once, see the derivative at a point calculator. For multivariable second partials, start with the partial derivative calculator and raise order or differentiate twice in the same variable.

Tips for clean higher-order work

Simplify after the first derivative when possible before reading the second pass in the step list. Watch product and quotient problems: most algebra errors appear on the second differentiation, not the first. If your course wants d²y/dx² for an implicit curve, find y′ with the implicit derivative calculator, then differentiate that expression carefully as a quotient.

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