Suppose z depends on x and y, and both x and y depend on t. Then z depends on t through two separate routes.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The rule
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt). One term per route, and the terms are added.
The tree diagram
Draw z branching to x and y, then each branching to t. Multiply along each branch and add the branches.
Why they add
Changing t nudges x and y at once, and each nudge moves z. Both effects happen, so both count.
Two parameters
If x and y depend on s and t, the same pattern gives ∂z/∂s and ∂z/∂t separately, one route at a time.
Implicit differentiation
For F(x, y) = 0 the chain rule gives dy/dx = −Fₓ/Fᵧ. The old technique falls out as a special case.
Every path of influence is added
If z depends on x and y, and both depend on t, then dz/dt sums the contribution through each. The multivariable chain rule adds one term per route from the input to the output.
A dependency tree organises it
Draw arrows from the final variable back to the independent one. Each complete path contributes a product of derivatives, and the rule is the sum over all paths.
It gives implicit differentiation
Applying the chain rule to F(x, y) = 0 produces dy/dx as a ratio of partial derivatives. The technique from single-variable calculus turns out to be a special case of this rule.
And multivariable related rates
A quantity depending on several changing measurements has a rate combining all of them. That is how error propagation and sensitivity analysis are computed in the sciences.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 2) · length 3.61
- Vector b(4, 1) · length 4.12
Step 3: Watch an Example
One step at a time.
Watch Tomas Follow Both Routes
Tomas has z with ∂z/∂x = 4 and ∂z/∂y = 3, while dx/dt = 2 and dy/dt = 5.
- Step 1
He takes the x route first: 4 × 2 = 8.
Step 4: Your Turn
Practice makes it stick.
The Heated Plate
Problem 1 of 2
∂T/∂x = 2 and ∂T/∂y = −1, with dx/dt = 3 and dy/dt = 4. What is dT/dt?
The Frozen Route
Problem 2 of 2
∂z/∂x = 5, ∂z/∂y = 9, dx/dt = 0 and dy/dt = 2. What is dz/dt?
Follow the Routes
1 of 8
∂z/∂x = 1, ∂z/∂y = 1, dx/dt = 6, dy/dt = 7. What is dz/dt?
2 of 8
∂z/∂x = 3, ∂z/∂y = 0, dx/dt = 4, dy/dt = 100. What is dz/dt?
3 of 8
How many terms does the chain rule have when z depends on three variables?
4 of 8
∂z/∂x = −2, ∂z/∂y = 4, dx/dt = 5, dy/dt = 3. What is dz/dt?
5 of 8
Fₓ = 6 and Fᵧ = 3 for F(x, y) = 0. What is dy/dx?
6 of 8
Are the chain rule terms added or multiplied together? Enter 1 for added, 2 for multiplied.
7 of 8
Order the steps of a chain rule computation.
- 1Multiply the derivatives along each route
- 2Add the routes together
- 3Substitute the values at the point
- 4Draw the tree of dependencies
8 of 8
∂z/∂x = 7 and dx/dt = 0, with no y dependence. What is dz/dt?
Step 5: Quick Check
Show what you know.
Question 1 of 2
∂z/∂x = 2, ∂z/∂y = 6, dx/dt = 3, dy/dt = 1. What is dz/dt?
Question 2 of 2
Why are the chain rule terms added?
What You Learned
- When z depends on x and y, and both depend on t, change travels along two routes.
- Multiply the derivatives along each route, then add the routes.
- Implicit differentiation is the same rule in disguise: dy/dx = −Fₓ/Fᵧ.