What is a partial derivative?
The derivative of a multivariable function with respect to one variable, holding others fixed.
Partial Derivative Calculator
Enter a general 2D quadratic polynomial and a point to evaluate both first partials.
Enter f(x,y) and the point
Provide coefficients a…f_const and coordinates (x, y).
About this calculator
How to use Partial Derivative Calculator, plus the formulas and logic behind the results.
Tool description
Partial derivatives treat one variable as active while holding the other fixed.
How to use
- Enter all six coefficients of the quadratic polynomial.
- Enter the evaluation point (x, y).
- Select Calculate to view ∂f/∂x, ∂f/∂y, and f(x,y).
- Combine results into ∇f if needed.
Formulas & reference
fₓ = 2ax + cy + dfᵧ = 2by + cx + e
Partial tips
- Hold y fixed when computing fₓ
- Hold x fixed when computing fᵧ
- Mixed partials fₓᵧ = fᵧₓ = c for this smooth polynomial
- Gradient is the vector (fₓ, fᵧ)
Calculation logic
Forms symbolic first partials of the quadratic and evaluates them at the given point.
Field name fconst is the constant term (PHP reserved-word friendly).
Partial Derivative Calculator FAQ
Common questions about this calculator and how to use the results.
What f is supported?
f(x,y)=ax²+by²+cxy+dx+ey+constant.
Why evaluate at a point?
Partials are functions; the tool also returns numeric values at (x,y).
Is the constant needed?
It affects f but not the first partials.
Can coefficients be zero?
Yes—omit unused terms by setting them to 0.
Is this free?
Yes.