A particle moves along a circle: x(t) = cos(t), y(t) = sin(t). The temperature field is T(x, y) = x² + y². What is dT/dt?
A0 — the particle moves along the level set x² + y² = 1, so T is constant
B−4xy sin(t)cos(t) — multiply the two chain contributions together
C−2cos(t)sin(t) — only the x-contribution matters
D2(cos t − sin t) — differentiate x and y and add them without partial derivatives
By the multivariable chain rule: dT/dt = (∂T/∂x)(dx/dt) + (∂T/∂y)(dy/dt) = (2x)(−sin t) + (2y)(cos t) = −2cos(t)sin(t) + 2sin(t)cos(t) = 0. The answer is zero because T = x² + y² = 1 everywhere on the unit circle — T is constant along this path. Option B is the critical misconception: the two chain contributions are added, not multiplied. Each intermediate variable contributes an independent additive term.
Question 2 Multiple Choice
For z = f(x, y) where x = x(s, t) and y = y(s, t), what is ∂z/∂s according to the multivariable chain rule?
A(∂f/∂x)(∂x/∂s) + (∂f/∂y)(∂y/∂s)
B(∂f/∂x)(∂x/∂s) · (∂f/∂y)(∂y/∂s)
C∂f/∂x + ∂f/∂y
D(∂f/∂x + ∂f/∂y)(∂x/∂s + ∂y/∂s)
The dependency diagram shows two paths from z to s: one through x and one through y. Each path contributes one term — multiply the derivatives along that path — and the two terms are added. This gives (∂f/∂x)(∂x/∂s) + (∂f/∂y)(∂y/∂s). Option B multiplies the two contributions, which would only be correct if the effects were compounding rather than simultaneous. Options C and D incorrectly separate the partial derivatives.
Question 3 True / False
When z depends on two intermediate variables x and y, each of which depends on t, the number of terms in dz/dt equals the number of intermediate variables (two).
TTrue
FFalse
Answer: True
True. The multivariable chain rule produces one term per path from z to the final variable t. With two intermediate variables, there are exactly two paths (z→x→t and z→y→t), giving two terms: (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt). This pattern generalizes: three intermediate variables give three terms, and so on. The dependency diagram makes this explicit — count the paths, and you have the number of terms.
Question 4 True / False
For z = f(x, y) with x = g(t) and y = h(t), the chain rule gives dz/dt = (∂f/∂x)(dx/dt) · (∂f/∂y)(dy/dt).
TTrue
FFalse
Answer: False
False. The correct formula is dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) — the contributions are added, not multiplied. Multiplication would apply if one effect depended on the other, but x and y change simultaneously and independently. Each is a separate channel through which changes in t reach z, so their contributions to z's total rate of change are summed.
Question 5 Short Answer
Explain why the multivariable chain rule adds contributions from each intermediate variable rather than multiplying them.
Think about your answer, then reveal below.
Model answer: Each intermediate variable is a separate, simultaneous channel through which a small change in t affects z. As t changes by Δt, x changes by (dx/dt)Δt and y changes by (dy/dt)Δt, and each of these changes independently nudges z. The x-channel contributes approximately (∂f/∂x)(dx/dt)Δt to z, and the y-channel contributes (∂f/∂y)(dy/dt)Δt. Since both effects happen at the same time through independent pathways, the total change in z is the sum. Multiplication would imply one effect depends on the other.
This is the key structural difference from the single-variable chain rule, which has only one intermediate variable and thus one term. The additive structure reflects the linearity of the first-order approximation (total differential): dz ≈ (∂f/∂x)dx + (∂f/∂y)dy. The chain rule is just this differential formula divided by dt.