A fourth degree polynomial
- Input
- d²/dx² (x^4 - 2x^2)
- Expected output
- 12x^2 - 4
Equals zero at x = ±1/sqrt(3), this function's two symmetric inflection points.
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second derivative calculator
The second derivative is the derivative of the derivative, and it measures how the function's slope is changing (its concavity). This page opens with order 2 already selected, using x⁴ - 2x² as the example, a function with two symmetric inflection points.
Equals zero at x = ±1/sqrt(3), this function's two symmetric inflection points.
Constant and always positive: the graph of x² is concave up across its whole domain, with no inflection point at all.
Polynomials, trigonometric functions (sine, cosine, tangent and their inverses), hyperbolic functions, exponentials and logarithms, including combinations built from sums, subtraction, products, quotients and composition (the chain rule). Factorial, the remainder operator (%) and min/max are not elementarily differentiable, and the calculator says so instead of risking a wrong result.
Just change the "Order of the derivative" field to the number you want (the calculator accepts 1 through 6); the step by step of each intermediate order appears separately, so you can follow how the expression changes at every successive differentiation.
Not necessarily, the second derivative's sign actually needs to switch from one side of that point to the other. The calculator checks this before listing an inflection point, so a zero where the concavity does not genuinely change (which can happen in rarer cases) does not show up in the list.
Examples by rule
4 * (3 * x^2) - 4
4 \cdot \left(3 \cdot x^{2}\right) - 4
No formula renderer on this page: copy the LaTeX above and paste it into an editor that displays it, like a document or a note.
Steps for the order 1 derivative
Difference rule: the derivative of u - v is u' - v'.
Power rule (with the chain rule): the derivative of u^n, with n constant, is n * u^(n-1) * u'.
Derivative of the variable with respect to itself: the result is 1.
Product rule: the derivative of u * v is u' * v + u * v'.
Derivative of a constant: the result is always 0.
Power rule (with the chain rule): the derivative of u^n, with n constant, is n * u^(n-1) * u'.
Derivative of the variable with respect to itself: the result is 1.
Steps for the order 2 derivative
Difference rule: the derivative of u - v is u' - v'.
Product rule: the derivative of u * v is u' * v + u * v'.
Derivative of a constant: the result is always 0.
Power rule (with the chain rule): the derivative of u^n, with n constant, is n * u^(n-1) * u'.
Derivative of the variable with respect to itself: the result is 1.
Product rule: the derivative of u * v is u' * v + u * v'.
Derivative of a constant: the result is always 0.
Product rule: the derivative of u * v is u' * v + u * v'.
Derivative of a constant: the result is always 0.
Derivative of the variable with respect to itself: the result is 1.
Everything runs in your browser: the expression you type, the symbolic differentiation and the graph never go through any server.