Enter a function of x, y and z, pick the variables to differentiate by, and get the exact partial derivative with every rule explained. Evaluate it at a point and see the full gradient and Hessian matrix.
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Examples:
Variables x, y, z (and t). Use ^ for powers, * or implicit multiplication (3xy), sqrt, ln, log, e^x, sin, cos, tan, asin, acos, atan, sinh, cosh, abs, pi.
Differentiate with respect to
Tap variables in order: x = ∂f/∂x, xx = ∂²f/∂x², xy = fxy (first x, then y).
Evaluate at a point (optional)
Result
Step-by-step
Gradient (all first partial derivatives)
Second partial derivatives (Hessian)
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How to use the Partial Derivative Calculator
Type the function using x, y and z, for example x^2*y^3 + 3xy or sin(x*y). Or tap an example.
Choose the derivative: tap ∂/∂x, ∂/∂y or ∂/∂z in the order you want, or a shortcut like fxx or fxy. The order box shows the sequence (xy means x first, then y).
Optionally enter a point such as x = 1, y = 2 to evaluate the derivative there.
Press Calculate partial derivative to see the result, the value at the point and the steps.
Scroll down for the gradient and Hessian, and copy the result with one click.
Rules for partial derivatives
Every single-variable rule still works. The only new idea is that the other variables behave like constants. Here is how each rule looks when differentiating with respect to x:
Rule
Pattern
Example f(x, y)
∂f/∂x
Constant
Terms without x → 0
5y³
0
Constant multiple
∂/∂x [c(y)·g(x)] = c(y)·g′(x)
y²·x³
3x²y²
Power
∂/∂x xⁿ = n·xⁿ⁻¹
x⁴ + y
4x³
Product
(uv)′ = u′v + uv′
x·sin(xy)
sin(xy) + xy·cos(xy)
Quotient
(u/v)′ = (u′v − uv′)/v²
x/(x + y)
y/(x + y)²
Chain
∂/∂x g(u) = g′(u)·∂u/∂x
ex²y
2xy·ex²y
Logarithm
∂/∂x ln(u) = (∂u/∂x)/u
ln(x² + y²)
2x/(x² + y²)
Notation cheat sheet
Subscript
Leibniz
Meaning
Enter as
fx
∂f/∂x
Rate of change in the x direction
x
fxx
∂²f/∂x²
Concavity in the x direction
xx
fxy
∂²f/∂y∂x
First x, then y
xy
fyx
∂²f/∂x∂y
First y, then x
yx
fxyz
∂³f/∂z∂y∂x
x, then y, then z
xyz
Worked example with a second-derivative test
Take f(x, y) = x³ − 3xy + y³. The first partials are fx = 3x² − 3y and fy = 3y² − 3x. Setting both to zero gives y = x² and x = y², so the critical points are (0, 0) and (1, 1). The second partials are fxx = 6x, fyy = 6y and fxy = −3, so D = 36xy − 9. At (0, 0), D = −9 < 0: a saddle point. At (1, 1), D = 27 > 0 and fxx = 6 > 0: a local minimum with f(1, 1) = −1. Type the function above and evaluate at each point to check every number.
Want to see the surface itself? Paste the same function into the 3D Graphing Calculator to rotate it and spot the saddle and the minimum.
Frequently asked questions
What is a partial derivative?
A partial derivative measures how a function of several variables changes when only one variable changes and all the others are held constant. For f(x, y) = x²y³, treating y as a constant gives ∂f/∂x = 2xy³, and treating x as a constant gives ∂f/∂y = 3x²y².
How do you calculate a partial derivative?
Pick the variable you differentiate with respect to, treat every other variable as if it were a number, and apply the ordinary derivative rules (power, product, quotient and chain rule). Terms that do not contain the variable are constants and differentiate to 0.
What does f_xy mean?
f_xy is a mixed second partial derivative: differentiate f with respect to x first, then differentiate the result with respect to y. In Leibniz notation it is written ∂²f/∂y∂x, with the variables read right to left. Enter it here as the sequence xy.
Is f_xy always equal to f_yx?
For functions whose second partial derivatives are continuous (true for nearly every formula you meet in class) the order does not matter, so f_xy = f_yx. This is Clairaut's (or Schwarz's) theorem. The Hessian section checks it for your function.
What is the gradient?
The gradient ∇f is the vector of all first partial derivatives, ⟨f_x, f_y, f_z⟩. At a point it points in the direction of steepest increase of f, and its length is the rate of increase in that direction.
What is the Hessian matrix used for?
The Hessian is the matrix of all second partial derivatives. At a critical point (where the gradient is zero) it classifies the point: for two variables, D = f_xx·f_yy − f_xy². D > 0 with f_xx > 0 is a local minimum, D > 0 with f_xx < 0 is a local maximum, and D < 0 is a saddle point.
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