Partial Derivative of Multivariable Function Calculator

Calculation Result

Metric	Value

Calculator

Function f(x,y,z)

Use explicit multiplication, such as x*y and 2*x.

x value

y value

z value

Step size h

Plot plane

Plot span

Grid size

dx for total differential

dy for total differential

dz for total differential

Plotly Graph

About This Calculator

This partial derivative of multivariable function calculator helps you study local change in a scalar function with two or three variables. Enter an expression, choose an evaluation point, and the tool estimates the first partial derivatives with respect to x, y, and z. It also computes second partial derivatives, mixed partials, gradient magnitude, and a total differential estimate.

These results are useful in multivariable calculus, optimization, machine learning, engineering analysis, and physics. Partial derivatives measure how the function changes when one variable moves and the others stay fixed. Mixed derivatives help you inspect curvature interactions between variables. The gradient magnitude summarizes the local steepness of the function at the chosen point.

The graph adds a local visual interpretation. You can inspect an XY, XZ, or YZ slice around the chosen point while one variable remains fixed. That makes it easier to connect numeric derivatives with nearby surface behavior. Export features let you save the computed values quickly for assignments, reports, or review notes.

Formula Used

For a function f(x,y,z) and a small step h: \partialf/\partialx \approx [f(x+h,y,z) - f(x-h,y,z)] / (2h) \partialf/\partialy \approx [f(x,y+h,z) - f(x,y-h,z)] / (2h) \partialf/\partialz \approx [f(x,y,z+h) - f(x,y,z-h)] / (2h) \partial²f/\partialx² \approx [f(x+h,y,z) - 2f(x,y,z) + f(x-h,y,z)] / h² \partial²f/\partialy² \approx [f(x,y+h,z) - 2f(x,y,z) + f(x,y-h,z)] / h² \partial²f/\partialz² \approx [f(x,y,z+h) - 2f(x,y,z) + f(x,y,z-h)] / h² \partial²f/\partialx\partialy \approx [f(x+h,y+h,z) - f(x+h,y-h,z) - f(x-h,y+h,z) + f(x-h,y-h,z)] / (4h²) \partial²f/\partialx\partialz \approx [f(x+h,y,z+h) - f(x+h,y,z-h) - f(x-h,y,z+h) + f(x-h,y,z-h)] / (4h²) \partial²f/\partialy\partialz \approx [f(x,y+h,z+h) - f(x,y+h,z-h) - f(x,y-h,z+h) + f(x,y-h,z-h)] / (4h²) Gradient magnitude: |\nablaf| = \sqrt[(\partialf/\partialx)² + (\partialf/\partialy)² + (\partialf/\partialz)²] Total differential estimate: df \approx (\partialf/\partialx)dx + (\partialf/\partialy)dy + (\partialf/\partialz)dz

How to Use This Calculator

Enter the multivariable expression in the function field.
Type the evaluation point values for x, y, and z.
Set a positive step size h for the finite difference estimate.
Select the plotting plane for the surface slice.
Choose a plot span and grid size.
Optionally enter dx, dy, and dz for total differential.
Press Calculate to show the result table above the form.
Use the export buttons to save the results as CSV or PDF.

Example Data Table

Function	Point	h	∂f/∂x	∂f/∂y	∂f/∂z
x^2y + sin(z) + xy*z	(1, 2, 0.5)	0.001	5	1.5	2.877582562

FAQs

What does this calculator compute?

It estimates first partial derivatives, second partial derivatives, mixed partials, gradient magnitude, function value, and total differential from one multivariable expression at a chosen point.

Does it work for two or three variables?

Yes. You can enter expressions using x, y, z, or any subset of them. If a variable is absent, its related derivative usually evaluates near zero.

Are the derivatives symbolic or numerical?

They are numerical estimates based on central difference formulas. Smaller step sizes often improve accuracy, but extremely tiny values can increase floating point error.

What syntax should I use for functions?

Use explicit multiplication and standard functions, such as x*y, x^2, sin(x), cos(y), exp(x), sqrt(x), and log(x). Constants pi and e are supported.

Why is the step size important?

The step size controls the spacing used in finite differences. A moderate positive value usually balances truncation error and rounding error for stable results.

What is the graph showing?

The graph displays a local surface slice of the expression on the selected plane. One variable stays fixed while the other two vary around your chosen point.

Can I export the results?

Yes. After calculation, you can download the result table as CSV or PDF for reports, homework checking, or documentation.

What is the total differential estimate?

It approximates the small change in the function using df ≈ fx·dx + fy·dy + fz·dz. This is useful for local sensitivity analysis.