Home

Partial Derivatives Calculator

Numerically compute first and second partial derivatives of f(x,y) at a point.

展开更多 ▾
f_x = ∂f/∂x ， f_y = ∂f/∂y
e.g. x^2*y + y^3

📖 Tutorial | Partial Derivatives

1. Definition
Partial derivatives f_x=∂f/∂x treat y as constant; f_y treats x as constant. Mixed derivatives f_xy=f_yx when continuous.
2. Symbols
f_xderivative w.r.t. x
f_yderivative w.r.t. y
f_xyx then y
(x₀,y₀)the point
3. How it works
  • Central differences for first order;
  • Difference the first derivatives for second order;
  • h balances truncation and round-off.
4. Steps
  1. Enter f(x,y);
  2. Enter (x₀,y₀);
  3. Click Calculate.
5. Example
Example: f=x²y+y³ at (2,1).
Solution: f_x=2xy=4, f_y=x²+3y²=7; f_xx=2, f_xy=4, f_yy=6.
6. Pitfalls
Hold the other variable constant;
Mixed derivatives are order-independent when continuous;
Numerical differences carry small errors.

❓ FAQ | Partial Derivatives

How is a partial derivative different?
Only one variable moves; the rest are fixed.
Are f_xy and f_yx equal?
Yes when continuous (Clairaut).
Why numerical differences?
No manual derivation needed.
What if f is not smooth?
The difference is inaccurate; use a smooth region.
Higher orders?
This tool gives first and second order.