# Chain Rule Measurements

Keywords: Multivariable Calculus, Rate of Change, Partial Derivative, Chain Rule, Surfaces

### Highlights of the activity

1. This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives.
2. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule.
3. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the $r$-direction and derivatives in $x$- and $y$-directions using the chain rule.

### Reasons to spend class time on the activity

After this activity, students will be able to

1. Measure partial derivatives in $x$-, $y$-, and $r$-direction on a three-dimensional surface;
2. Express a partial derivative in any arbitrary direction in terms of those in $x$-, $y$-directions;
3. Verify the chain rule relation $\frac{\partial f}{\partial r}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial r}$ numerically by comparing their measured values.

### Student Handouts

Authors: (c) Aaron Wangberg and the "Raising Calculus to the Surface" team; used by permission.

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