Calculus
Partial Derivatives Calculator
Estimate ∂f/∂x and ∂f/∂y at a specific (x, y) coordinate and inspect the gradient magnitude.
Estimate ∂f/∂x and ∂f/∂y for a multivariable function.
Central difference in multiple dimensions
∂f/∂x ≈ [f(x + h, y) − f(x − h, y)] / (2h)
The same stencil applies to ∂f/∂y, giving a quick view of how the surface slopes along each axis.
How to use
- Enter a function f(x, y) using x and y variables.
- Specify the evaluation point (x₀, y₀) and the finite-difference step size.
- Read off partial derivatives and the gradient magnitude.
Example
Input: f(x, y) = x^2 * y + sin(y) at (1, 0.5)
Output: ∂f/∂x ≈ 1.0, ∂f/∂y ≈ 1.5403, ‖∇f‖ ≈ 1.835
Student-friendly breakdown
This walkthrough emphasizes the most searched ideas for Partial Derivatives Calculator: partial derivative calculator, gradient calculator, multivariable derivative calculator, partial derivative calculator with steps. Start with the formula above, then follow the guided steps to double-check your work. For quick revision, highlight the givens, plug into the equation, and finish by verifying your units.
Need more support? Use the links below to open the long-form guide, browse additional examples, or hop into adjacent calculators within the same topic — each one is a quick way to double-check your work or handle a related question without starting from scratch.
Deep dive & study plan
Partial Derivatives Calculator: Computes partial derivatives for functions of two variables. It's built around partial derivative, gradient, multivariable, so you can go from a raw question to a checked answer without switching tools.
The math behind it: The same stencil applies to ∂f/∂y, giving a quick view of how the surface slopes along each axis. The core relationship is ∂f/∂x ≈ [f(x + h, y) − f(x − h, y)] / (2h), shown above the calculator so you can see exactly how your inputs turn into the result.
To use it well: (1) Enter a function f(x, y) using x and y variables. (2) Specify the evaluation point (x₀, y₀) and the finite-difference step size. (3) Read off partial derivatives and the gradient magnitude. Keep your units consistent as you go, and re-run a case you already know the answer to — it's the fastest way to catch a typo before it throws off a result you're relying on.
Worked example: entering f(x, y) = x^2 * y + sin(y) at (1, 0.5) returns ∂f/∂x ≈ 1.0, ∂f/∂y ≈ 1.5403, ‖∇f‖ ≈ 1.835. Try swapping in your own numbers next, especially a case you're unsure about, before you use this for something that matters.
Quick retention checklist
- Speak the formula aloud (or annotate it) so the relationships stick.
- Write each step in your own words and compare with the numbered list above.
- Swap in new numbers for the Example to make sure the calculator (and your logic) handles edge cases.
- Check at least one related calculator below — it's the fastest way to confirm your numbers still line up from a different angle.
FAQ & notes
Can I extend this to more variables?
This panel is tuned for two variables. For higher dimensions, repeat the approach by holding other variables constant.
What step size should I use?
The default (1e-4) balances precision and stability. Reduce it for well-behaved functions or increase it if round-off dominates.
What formula does the Partial Derivatives Calculator use?
The same stencil applies to ∂f/∂y, giving a quick view of how the surface slopes along each axis.
How do I use the Partial Derivatives Calculator?
Enter a function f(x, y) using x and y variables. Specify the evaluation point (x₀, y₀) and the finite-difference step size. Read off partial derivatives and the gradient magnitude.