Educational tool; results are estimates for learning, verify with your textbook or instructor.
This page is a Jacobian matrix calculator with steps. You type one, two, or three formulas in the inputs x, y, and z. The page differentiates each formula with respect to each input that is in use, stacks those partial derivatives into a matrix, and writes the determinant when the matrix is square. Leave the point blank for the symbolic matrix. Fill every coordinate and the page also evaluates the matrix there.
A Jacobian is the derivative of a map that may have several outputs. One output is a gradient written as a row. Several outputs are those rows stacked. The determinant, when it exists, is the signed volume scale of that linear approximation. It is not a second name for the matrix. The story of that distinction is Jacobian matrices in plain language. The course map is multivariable and vector calculus.
The box opens on (x^2 − y^2, 2xy), evaluated at (1, 0). The partial derivatives are 2x, −2y, 2y, and 2x. The determinant is 4x^2 + 4y^2, which equals 4 at that point. The same lines are printed below, so you can read them with scripts turned off. A numerical gradient of one scalar formula, by central differences, is the gradient and directional derivative calculator. This page does the algebra instead.
How the partial derivatives are built
Each partial derivative holds the other inputs fixed and differentiates with respect to one letter. A sum is differentiated term by term. A product uses the product rule. A power uses the power rule, and a function of a formula uses the chain rule. Like terms are combined before the matrix is printed, so 2x + 0 is printed as 2x.
The grammar is small on purpose. Juxtaposition means multiplication, and it binds tighter than the slash and the star. That is why 2xy is 2 times x times y, and why 1/2x is 1 divided by 2x. Write 1/2*x when you want half of x. A leading minus sits outside a power, so −x^2 is the negative of x squared, which is the usual calculus convention. The letter e is refused. Write exp(x). The name log is the natural logarithm, the same function as ln. Integer powers run from −12 to 12. A small positive power of a sum is expanded. A negative power of a sum of several terms stays in parentheses.
When the number of outputs equals the number of inputs, the page expands the determinant. A 1 by 1 determinant is the ordinary derivative. A 2 by 2 determinant is ad − bc. A 3 by 3 determinant is the usual cofactor expansion. A rectangular matrix has no determinant, and the page says so instead of inventing one. You can still paste a numeric 2 by 2 matrix into the 2 by 2 matrix calculator if you want the inverse of a Jacobian you have already evaluated.
A fully worked example
Let F(x, y) = (x^2 − y^2, 2xy), and evaluate at (1, 0). The first component has partial derivatives 2x and −2y. The second has partial derivatives 2y and 2x. The matrix is therefore the rows (2x, −2y) and (2y, 2x). Its determinant is 4x^2 + 4y^2. At (1, 0) the matrix is the rows (2, 0) and (0, 2), and the determinant is 4. That factor 4 is the local area scale: a small region near (1, 0) is stretched by 4, and the orientation stays the same because the determinant is positive. The lines below match the calculator.
Inputs: x, y.
Each partial derivative holds the other inputs fixed. Like terms are combined. A product uses the product rule, and a composition uses the chain rule.
Output 1 is x^2 − y^2.
Partial of output 1 with respect to x is 2x.
Partial of output 1 with respect to y is −2y.
Output 2 is 2xy.
Partial of output 2 with respect to x is 2y.
Partial of output 2 with respect to y is 2x.
Jacobian matrix, one output per row and one input per column.
Row 1: 2x, −2y
Row 2: 2y, 2x
Determinant 4x^2 + 4y^2.
For a square map the determinant is the signed volume scale of the linear approximation. It is not the same object as the matrix.
Evaluated at (1, 0).
Row 1 at the point: 2, 0
Row 2 at the point: 0, 2
Determinant at the point: 4.
Common mistakes
The first mistake is calling the determinant the Jacobian when the sentence needed the matrix, or the other way around. Invertibility, eigenvalues, and the chain rule use the matrix. A change of variables in an integral uses the absolute value of the determinant. OpenStax states that factor in the section cited below.
The second mistake is differentiating 2xy as if the letters were separate terms. The product rule on 2 · x · y, holding y fixed, gives 2y. Holding x fixed gives 2x. A missing factor of 2 usually means one of those letters was treated as a constant too early.
The third mistake is reading a printed whole number as a proof that the value is exact. sin, cos, tan, exp, ln, and sqrt are evaluated numerically at a point. A fraction with no such function is exact. The page says which case you are in.
The fourth mistake is using this page for a directional derivative of one scalar function. That object is a gradient dotted with a unit vector. Estimate it on the gradient calculator. This page will still give you the gradient row if you type a single output, and it will not divide by a direction you did not enter.
Pitfall. A zero determinant does not mean the derivative failed to exist. It means the linear approximation crushes some direction. The same map, at (0, 0), has the zero matrix, and that derivative is correct. The explainer works that example. This calculator will print the zero matrix if you move the point to the origin.
Related reading
The surrounding map is multivariable and vector calculus. The ordered linear algebra that makes a matrix feel like a machine is the linear algebra first course. Glossary lines for Jacobian, partial derivative, and gradient are the words this arithmetic is about.
After the matrix is numeric, the 2 by 2 matrix calculator finds a determinant or an inverse, and the 2 by 2 eigenvalue calculator finds stretch factors. Neither of those pages differentiates a formula. The elimination factors of a numeric matrix, which are not a Jacobian, are on the LU decomposition calculator.
Last reviewed: 26 October 2026.
