|Gradients and Directional Derivatives|
A partial derivative is the derivative of a several-variable function with respect to only one variable. This definition means that when we evaluate the partial derivative of a function with respect to a particular variable, all of the other input variables of the function are treated as constants. The partial derivative tells us how much a function is changing in the direction of a particular variable.
The partial derivative of a function f with respect to x is denoted .
- We take the partial derivative of the function with respect to y:
- Note that the terms and are treated as constants since they do not contain a y-factor, so go to zero when differentiated. Only the middle term is not constant with respect to y, so the coefficient of -2 remains after differentiation.
- Another example: We take the partial derivative of the function with respect to x:
- Again, notice that all variables except x are treated as constants.
Graphical Interpretation, in three dimensions
The partial derivative in three dimensions can be thought of as taking a "slice" of a function from and finding the derivative of the curve within this slice. Taking a slice means we take a plane along which one of the input variables is constant, find the curve contained by this plane, then differentiate this curve with respect to the other input variable.