Partial Derivative Calculator
Numeric partial derivatives of multivariable functions at a point, including gradients.
Enter your values
Enter the function f(x, y, z) you are working with.
Enter the x = you are working with.
Enter the y = you are working with.
Enter the z = (optional) you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- f at the point
- 2.909297
- ∂f/∂x
- 3.167706
- ∂f/∂y
- 0.583853
- ∂f/∂z
- 0
- Gradient ∇f
- (3.1677, 0.5839, 0)
- ∂²f/∂x²
- 0.36281
- ∂²f/∂y²
- -0.9093
Working
- 1Hold every variable except one constant.
- 2Differentiate with respect to the remaining variable (done here with a high-accuracy central difference).
- 3Collect the partials into the gradient vector ∇f.
Quick answer
A partial derivative measures how a multivariable function changes when one variable moves and the others stay fixed. This calculator returns ∂f/∂x, ∂f/∂y and ∂f/∂z at a chosen point plus the gradient vector.
∂f/∂x = lim h→0 [f(x+h, y) − f(x, y)] / h
At a glance
| What it does | Numeric partial derivatives of multivariable functions at a point, including gradients. |
|---|---|
| Category | Math |
| Inputs needed | Function f(x, y, z), x =, y =, z = (optional) |
| Main output | f at the point |
| Formula | ∂f/∂x = lim h→0 [f(x+h, y) − f(x, y)] / h |
| Cost | Free — no sign-up, no download |
How to use the Partial Derivative Calculator
- 1Type the function using x, y and z.
- 2Set the point where you want the derivative.
- 3Read ∂f/∂x, ∂f/∂y, ∂f/∂z and the gradient.
Inputs explained
- Function f(x, y, z)
- Enter the function f(x, y, z) you are working with.
- x =
- Enter the x = you are working with.
- y =
- Enter the y = you are working with.
- z = (optional)
- Enter the z = (optional) you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Function f(x, y, z)x^2*y + sin(x*y)
- x =1
- y =2
- z = (optional)0
Results
- f at the point2.909297
- ∂f/∂x3.167706
- ∂f/∂y0.583853
- ∂f/∂z0
- Gradient ∇f(3.1677, 0.5839, 0)
- ∂²f/∂x²0.36281
- ∂²f/∂y²-0.9093
- Hold every variable except one constant.
- Differentiate with respect to the remaining variable (done here with a high-accuracy central difference).
- Collect the partials into the gradient vector ∇f.
Frequently asked questions
What is a partial derivative?
It is the rate of change of a multivariable function with respect to one variable while all other variables are held constant.
How do you find the gradient?
Collect the partial derivatives into a vector: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z). The gradient points in the direction of steepest increase.
Is the result exact?
Values are computed numerically with a central-difference scheme, accurate to roughly 6 significant digits for smooth functions.
What do I need to enter into the Partial Derivative Calculator?
Just 4 values: function f(x, y, z), x =, y =, z = (optional). Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Partial Derivative Calculator work out the answer?
A partial derivative measures how a multivariable function changes when one variable moves and the others stay fixed. This calculator returns ∂f/∂x, ∂f/∂y and ∂f/∂z at a chosen point plus the gradient vector. It applies the formula ∂f/∂x = lim h→0 [f(x+h, y) − f(x, y)] / h and shows the working so you can check each step by hand.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.