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Cogito

Multivariable Calculus · Chapter 4 · Lesson 3

Practice Worksheet

The Multivariable Chain Rule

Every path the change can travel gets added up.

12 problems · about 24 minutes · F-IF.B.6, F-BF.A.1

NameDate

Figure — use these to answer the problems

One vector is the pair of partials and the other is the pair of rates. The chain rule is their dot product.
  • Vector a(3, 2) · length 3.61
  • Vector b(4, 1) · length 4.12
partial of z by x3
partial of z by y2
dx/dt4
dy/dt1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. ∂z/∂x = 2, ∂z/∂y = 6, dx/dt = 3, dy/dt = 1. What is dz/dt?

    Answer

  2. Why are the chain rule terms added?

    • Both variables change at once, so both effects on z happen together.
    • It is a convention with no particular reason.

  3. ∂z/∂x = 1, ∂z/∂y = 1, dx/dt = 6, dy/dt = 7. What is dz/dt?

    Answer

  4. ∂z/∂x = 3, ∂z/∂y = 0, dx/dt = 4, dy/dt = 100. What is dz/dt?

    Answer

  5. How many terms does the chain rule have when z depends on three variables?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. ∂z/∂x = −2, ∂z/∂y = 4, dx/dt = 5, dy/dt = 3. What is dz/dt?

    Answer

  2. Fₓ = 6 and Fᵧ = 3 for F(x, y) = 0. What is dy/dx?

    Answer

  3. Are the chain rule terms added or multiplied together? Enter 1 for added, 2 for multiplied.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Order the steps of a chain rule computation.

    Write 1 to 4 in the boxes to put these in order.

    • Draw the tree of dependencies
    • Multiply the derivatives along each route
    • Add the routes together
    • Substitute the values at the point

  2. ∂z/∂x = 7 and dx/dt = 0, with no y dependence. What is dz/dt?

    Answer

  3. The Heated Plate

    ∂T/∂x = 2 and ∂T/∂y = −1, with dx/dt = 3 and dy/dt = 4. What is dT/dt?

    Answer

  4. The Frozen Route

    ∂z/∂x = 5, ∂z/∂y = 9, dx/dt = 0 and dy/dt = 2. What is dz/dt?

    Answer