Step by step
The question
Question: find dz/dt when z = x² + xy, x = t² and y = t.
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.
x²: the power 2 comes down.
The power goes down by 1.
So 2x.
In xy, y is fixed and waits in front.
x gives 1.
So y.
So ∂z/∂x = 2x + y.
Term by term.
x² has no y: it is a fixed number, so it gives 0.
In xy, x is fixed and waits in front.
y gives 1.
So x.
Drop the zeros: ∂z/∂y = x.
x = t², so dx/dt = 2t.
y = t, so dy/dt = 1.
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 = t² and y = t in.
Each x becomes t² and each y becomes t.
Multiply each part out: numbers with numbers, and add the powers of t.
Add the like terms: dz/dt = 4t³ + 3t².
5 Check
Let's check another way.
Put x and y in first: z = t⁴ + t³.
Its derivative is 4t³ + 3t².
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.