Step by step
The question
Question: find dz/dt when z = x²y − y², x = 2t and y = t³.
1 Draw the tree
Draw a tree.
z depends on x and y.
Each of them depends on t.
Each branch has its own slope.
To go from z down to t, there are two paths.
Multiply along each path, then add the paths.
2 The pieces
Term by term.
In x²y, y is fixed and waits in front.
x² gives 2x.
So 2xy.
y² has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂z/∂x = 2xy.
Term by term.
In x²y, x² is fixed and waits in front.
y gives 1.
So x².
y²: the power 2 comes down: −1 times 2 is −2.
The power goes down by 1.
So 2y.
So ∂z/∂y = x² − 2y.
x = 2t, so dx/dt = 2.
y = t³, so dy/dt = 3t².
3 Use the chain rule
The chain rule: the x path plus the y path.
Put the four pieces in.
Careful!
Many people use only one path.
z changes through x and through y, so both paths count.
4 Write it in t
The answer should be in t only.
So put x = 2t and y = t³ in.
Each x becomes 2t and each y becomes t³.
Multiply each part out: numbers with numbers, and add the powers of t.
Add the like terms: dz/dt = 20t⁴ − 6t⁵.
5 Check
Let's check another way.
Put x and y in first: z = 4t⁵ − t⁶.
Its derivative is 20t⁴ − 6t⁵.
The same answer.
Right!
Let's look back.
First: draw the tree.
Then: the pieces.
Then: use the chain rule.
Then: write it in t.
Last: check.