A Partial Derivative Calculator is designed for functions with more than one independent variable. It finds the rate of change with respect to one variable while the other independent variables are held constant.
Understanding Partial Derivatives
For a function such as f(x,y), the partial derivative with respect to x is written βf/βx. During that calculation, y is treated as a constant. The partial derivative with respect to y follows the same idea with x held constant.
How to Use the Tool
Enter the multivariable function and select or provide the variable with respect to which you want to differentiate. Check the expression carefully because changing the selected variable changes the meaning of the result.
Example
Suppose f(x,y) = xΒ²y + 3y. With respect to x, y is constant, so βf/βx = 2xy. With respect to y, x is constant, giving βf/βy = xΒ² + 3. The two derivatives describe different directions of change.
Why It Is Useful
Partial derivatives are used in multivariable calculus, optimization, physics, engineering, and mathematical modeling. They are also the building blocks of the gradient, which combines the first partial derivatives into a vector describing the direction of greatest increase.
Reviewing the Result
A simple check is to temporarily treat the other variables as constants and differentiate the expression by hand. If a supposedly constant variable has been differentiated, the setup needs to be corrected.