SME
Calculus III (Multivariable) · Problem type
The chain rule for z = f(x, y)
Use this for “At (1, 2), ∂z/∂x = 4 and ∂z/∂y = 9. If x = 1 + 2t and y = 2 − t, find dz/dt at t = 0.”
Diagram
Each path from z down to t multiplies its branches; dz/dt adds the paths.
dz/dt
dz/dt = ∂z/∂x × dx/dt + ∂z/∂y × dy/dt
dz/dt = 4 × 2 + 9 × (−1) = −1
This is an example. Type your own values to start a new problem.
dz
dt
=
∂z
∂x
·
dx
dt
+
∂z
∂y
·
dy
dt
dz/dt
=
∂z/∂x
·
dx/dt
+
∂z/∂y
·
(
dy/dt
)
Formulas
dz/dt = (∂z/∂x) × (dx/dt) + (∂z/∂y) × (dy/dt)
−1 = 4 × 2 + 9 × (−1)
Assumptions
•
z depends on x and y, and x and y each depend on t.
•
Along each path of the tree, multiply the rates: ∂z/∂x × dx/dt, then ∂z/∂y × dy/dt.
•
Each path is one way a change in t reaches z, so dz/dt adds the two paths.
Step-by-step
WE KNOW
∂z/∂x = 4, ∂z/∂y = 9, dx/dt = 2, dy/dt = −1
FIND
rate of z in t (dz/dt)
Step 1 · Find rate of z in t (dz/dt)
Use dz/dt = (∂z/∂x) × (dx/dt) + (∂z/∂y) × (dy/dt)
Multiply along each path from z down to t, then add the two paths.
dz/dt = 4 × 2 + 9 × (−1)
dz/dt = 8 + (−9)
dz/dt = −1
Check
Put the numbers back in. Both sides should match.
−1 = 4 × 2 + 9 × (−1) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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