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# Chain Rule Measurements

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

### Highlights of the activity

- This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives.
- Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule.
- 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

- Measure partial derivatives in $x$-, $y$-, and $r$-direction on a three-dimensional surface;
- Express a partial derivative in any arbitrary direction in terms of those in $x$-, $y$-directions;
- 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.

### Reflections

### Instructor's Guide

### Student Handouts

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

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