Partial Derivative Calculator

Find partial derivatives of f(x, y) or f(x, y, z) up to second order, check Euler's theorem for homogeneous functions, and work out Jacobians, with every step checked.

Find	Partial derivatives (first and second order)
f(x, y) or f(x, y, z) =	x^2 y + sin(x y)
At the point (optional)

Find

f(x, y) or f(x, y, z) =

Use x, y and z. Write xy for x times y; e^(xy), sqrt, ln, sin and the rest work.

At the point (optional)

Values of the variables, in order, to evaluate the results there.

∂f/∂x, ∂f/∂y

2x·y + y·cos(x·y), x^2 + x·cos(x·y)

The mixed partial derivatives f_xy agree with the reversed order, as Clairaut's theorem says.

Exact form. Checked numerically.

Method: Differentiation rules, one variable at a time
Worked out with: AllCalcHub expression engine
Check: Every step compared at several values of x

∂f/∂x: 2x·y + y·cos(x·y)
∂f/∂y: x^2 + x·cos(x·y)
f_xx: 2y − y^2·sin(x·y)
f_xy: 2x + cos(x·y) − x·y·sin(x·y)
f_yy: −x^2·sin(x·y)

About the Partial Derivative Calculator

A partial derivative ∂f/∂x measures how f changes when only x changes, with the other variables held constant. The mixed partials f_xy and f_yx are equal whenever they are continuous (Clairaut's theorem), which this calculator confirms. The gradient ∇f = (f_x, f_y, f_z) points in the direction f increases fastest.

Euler's theorem: if f is homogeneous of degree n, meaning f(tx, ty) = tⁿ f(x, y), then x f_x + y f_y = n f, and x² f_xx + 2xy f_xy + y² f_yy = n(n − 1) f. The Jacobian ∂(u, v)/∂(x, y) is the determinant of the partial derivatives; it is the area scale factor in a change of variables (r for polar coordinates) and is 0 exactly when u and v are functionally dependent.

The formula

x ∂f/∂x + y ∂f/∂y = n f; J = ∂(u, v)/∂(x, y) = u_x v_y − u_y v_x

Homogeneous of degree n: f(tx, ty) = tⁿ f(x, y) for every t > 0
Polar coordinates: x = r cos θ, y = r sin θ gives J = r
Spherical coordinates: J = r² sin θ

Worked example

f = x² y + sin(xy)

f_x = 2xy + y cos(xy), f_y = x² + x cos(xy)
f_xy = f_yx = 2x + cos(xy) − xy sin(xy)

Questions people ask

How do I enter x times y?

Type xy or x*y. For three variables use x, y and z.

Why simplify the Jacobian with a symbolic engine?

Expanding the determinant leaves terms like r cos²θ + r sin²θ; the engine simplifies it to r, and the result is checked against the unsimplified determinant at several points.