Partial Derivative Calculator
Numeric partial derivatives of multivariable functions at a point, including gradients.
Enter your values
ⓘ Live results
- 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
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.
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.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.