Partial Derivatives Calculator
Enter f(x,y) and a point (x₀,y₀) to compute ∂f/∂x, ∂f/∂y and the mixed partial numerically.
Input Data
Results
At a glance:The partial ∂f/∂x differentiates with respect to x holding y constant (rate along x); ∂f/∂y is analogous. The mixed ∂²f/∂x∂y differentiates w.r.t. y then x (equal to the reverse for smooth functions). The gradient ∇f=(∂f/∂x, ∂f/∂y) points to steepest ascent.
Formula
Numeric: ∂f/∂x ≈ [f(x+h,y)−f(x−h,y)]/(2h).
∂f/∂y ≈ [f(x,y+h)−f(x,y−h)]/(2h).
Mixed: ∂²f/∂x∂y ≈ [f(x+h,y+h)−f(x+h,y−h)−f(x−h,y+h)+f(x−h,y−h)]/(4h²).
Gradient: ∇f = (∂f/∂x, ∂f/∂y).
$$\frac{\partial f}{\partial x} = \lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}$$$$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$$How to Use
- Enter f(x,y) (x, y, + - * / ^, sin/cos/exp/ln/sqrt, e, pi).
- Enter the point (x₀, y₀).
- The tool gives ∂f/∂x, ∂f/∂y, the mixed partial and ∂²f/∂x² (numeric).
- Case studies show analytic partials to compare with numeric results.
Basic two-variable partials
| f(x,y) | ∂f/∂x | ∂f/∂y |
|---|---|---|
| x²+y² | 2x | 2y |
| x·y | y | x |
| eˣsin y | eˣ sin y | eˣ cos y |
| x³y² | 3x²y² | 2x³y |
| ln(x²+y²) | 2x/(x²+y²) | 2y/(x²+y²) |
Numeric results may have small float error; read integer/simple values exactly.
Case Studies
f(x,y)=x²+y² at (1,2)
∂f/∂x = 2x → 2; ∂f/∂y = 2y → 4.
Mixed ∂²f/∂x∂y = 0 (no cross term).
f(x,y)=x·y
∂f/∂x = y; ∂f/∂y = x.
Mixed ∂²f/∂x∂y = 1 (order-independent).
f(x,y)=eˣ·sin y at (0,0)
∂f/∂x = eˣ sin y → 0; ∂f/∂y = eˣ cos y → 1.
Gradient ∇f = (0, 1).
f(x,y)=x³y²
∂f/∂x = 3x²y²; ∂f/∂y = 2x³y.
∂²f/∂x² = 6xy².
f(x,y)=ln(x²+y²)
∂f/∂x = 2x/(x²+y²); ∂f/∂y = 2y/(x²+y²).
At (1,1): ∂f/∂x = ∂f/∂y = 1.
Gradient and direction
∇f points to steepest ascent; its magnitude is the max rate.
For x²+y² at (1,2), ∇f=(2,4), magnitude √20≈4.472.
FAQ
Partial vs ordinary derivative?
A partial derivative differentiates with respect to one variable while holding others fixed, measuring a single-direction rate; ordinary derivatives are for one variable.
Does mixed-partial order matter?
If the function is twice continuously differentiable, ∂²f/∂x∂y = ∂²f/∂y∂x (Clairaut's theorem), independent of order.
What is the gradient?
The gradient ∇f = (∂f/∂x, ∂f/∂y) is the vector of partials, pointing in the direction of steepest increase, with magnitude the max rate.
Why small errors?
Central differences use tiny h, introducing rounding error; read integer/simple-fraction expectations exactly.
Which functions are supported?
x, y, arithmetic, ^, parentheses, sin/cos/tan/exp/ln/sqrt/abs with constants e, pi.
Three variables?
This version is limited to two variables f(x,y); three variables need extra variables and difference dimensions.
What is ∂²f/∂x²?
The second partial w.r.t. x, describing concavity along x; physically e.g. the spatial second derivative in heat equations.
Relation to total differential?
df = (∂f/∂x)dx + (∂f/∂y)dy; the partials are its coefficients, used for linear approximation and error propagation.
Related Tools
References
Content review: Calculatorism Science Team. Central-difference partial and mixed derivatives verified. Results are for reference only.