Prerequisite concepts
One can view $\frac{df}{dx}$ as approximately given by a fraction where $df$ is a small change in $f$ and $dx$ is a small change in $x$.
When taking a derivative of a multivariable function, it is possible to consider a simpler situation in which some of the variables are considered to be constant.
Taking derivatives of the same function with respect to different variables will produce different results. In physics, these different results might have different dimensions and thus the derivative can have wildly different physical interpretations based off of "'with respect to' what".
The Heater II
This small group activity is designed to help students interpret partial derivatives using contour diagrams. Students work in small groups to determine rates of change using a contour diagram showing isotherms over time and space. The whole class wrap-up discussion emphasizes giving a physical interpretation for the derivative, the value of units in thinking about functions and derivatives, and the need to specify “with respect to what” when finding derivatives in multivariable contexts. FIXME: Add Magnitude of partial derivative/contour map pair once made
Representations used
Concepts taught
Derivatives of multivariable functions can be found by holding one or more variables constant (subject to physical limitations).
For a smooth, two dimensional function, a partial derivative can be found at any point and in any direction.