Partial Derivatives
Partial derivatives and gradient of f(x,y)
About this calculator
The Partial Derivatives Calculator differentiates f(x, y) symbolically, producing exact formulas for ∂f/∂x and ∂f/∂y plus the second-order partials fxx, fyy, and the mixed fxy. Given a point (x₀, y₀) it also evaluates f, the gradient ∇f, its magnitude |∇f|, and the direction of steepest ascent with 30-digit decimal precision. Non-smooth functions such as abs or floor are rejected rather than mis-differentiated, and undefined evaluation points are flagged. Useful for multivariable calculus, optimization, and machine-learning study.
How to use the Partial Derivatives calculator
- Enter the values for your problem into the input fields.
- Read the result — it updates instantly as you type.
- Check the formula and the visual explanation to follow how the answer was found.
- Copy the page URL to share the exact calculation.
Common examples
- f = x² + y² → fx = 2x, fy = 2y, fxx = fyy = 2; at (1, 2): ∇f = (2, 4), |∇f| ≈ 4.4721
- f = x²y + sin(y) → fx = 2xy, fy = x² + cos(y), mixed partial fxy = 2x
- f = exp(x)·cos(y) at (0, 0) → f = 1, fx = 1, fy = 0, steepest ascent along +x (0°)
Frequently asked questions
How does the Partial Derivatives calculator work?
Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.
When would I use the Partial Derivatives calculator?
It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.
How accurate are the results?
Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.