z = x^2 y, x = t^2, y = 3t, find dz/dt Easy

The question

Question: find dz/dt when z = x²y, x = t² and y = 3t.

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.

So ∂z/∂x = 2xy.

Term by term.

In x²y, x² is fixed and waits in front.

y gives 1.

So x².

So ∂z/∂y = x².

x = t², so dx/dt = 2t.

y = 3t, so dy/dt = 3.

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 = 3t in.

Each x becomes t² and each y becomes 3t.

Multiply each part out: numbers with numbers, and add the powers of t.

Add the like terms: dz/dt = 15t⁴.

5 Check

Let's check another way.

Put x and y in first: z = 3t⁵.

Its derivative is 15t⁴.

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.

▶ Watch it solved

z = x^2 + xy, x = t^2, y = t, find dz/dt Medium

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.

▶ Watch it solved

z = x^2 y - y^2, x = 2t, y = t^3, find dz/dt Harder

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.

▶ Watch it solved