* for multiplication, ^ for powers, and functions like sin(), exp(), ln().What this partial differentiation calculator does
This tool computes a partial derivative of a multivariable function with respect to one chosen variable.
You can also request higher-order derivatives and evaluate the result at a specific point.
For example, if your function is f(x,y)=x^2y+sin(xy), you can differentiate with respect to x and then evaluate at x=1, y=2.
How to use it
1) Enter your function
Type your expression using standard calculator syntax:
- Multiplication:
x*y - Powers:
x^3 - Trig and exponential:
sin(x),cos(y),exp(z) - Logarithm:
ln(x)
2) Choose the variable
In partial derivatives, all other variables are treated as constants.
So ∂/∂x means only x changes.
3) Set derivative order
Keep order = 1 for first partial derivative.
Set order = 2 for a second derivative with respect to the same variable (for example, ∂²f/∂x²).
4) Optional: evaluate at a point
Provide values like x=2, y=3.
The calculator will show the symbolic derivative and its numeric value at that point.
Worked example
Suppose f(x,y)=x^2y+sin(xy).
Differentiate with respect to x:
∂/∂x (x^2y) = 2xy∂/∂x (sin(xy)) = cos(xy)·y(chain rule)
So, ∂f/∂x = 2xy + y*cos(xy).
At (x,y)=(1,2), this becomes 4 + 2*cos(2).
Common input mistakes and fixes
- Missing multiplication sign: write
x*y, not justxy. - Invalid variable assignment: use
x=1, y=2format. - Order out of range: keep it as a positive integer (1 to 10 here).
- Wrong function names: use
sin,cos,tan,exp,ln.
Why partial derivatives matter
Partial derivatives are central in many fields:
- Machine learning: gradients for model optimization.
- Economics: marginal change with respect to one input.
- Physics: heat flow, wave equations, and field models.
- Engineering: sensitivity analysis and system design.
Quick tips
- Start symbolic first, then evaluate numerically.
- Use higher-order derivatives to study curvature and stability.
- Check units when applying derivatives to real-world models.