Partial derivative

This is a Helper Page for:
Divergence Theorem
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 $\frac{\partial{f}}{\partial{x}}$.

Examples

• We take the partial derivative of the function $f(x,y,z) = x^2 -2y + \sin(z)$ with respect to y:

$\frac{\partial{f}}{\partial{y}} = -2$

Note that the terms $x^2$ and $\sin(z)$ 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 $f(x,y,z) = x^4y -2yz +e^{xyz}$ with respect to x:

$\frac{\partial{f}}{\partial{x}} = 4yx^3+yze^{xyz}$

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 $\mathbb{R}^2 \rightarrow \mathbb{R}$ 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.

For the graph of a function (blue) the partial derivative can be thought of as taking a plane (purple), finding the curve that the plane contains (yellow), then taking the derivative of this curve.