Step by step
The question
Question: find the partial derivatives of f(x, y, z) = x²y + yz³.
1 Partial by x
The partial derivative ∂f/∂x is the slope when only x changes.
For ∂f/∂x, only x changes.
y and z stay fixed: they act like plain numbers.
They are purple now.
Term by term.
In x²y, y is fixed and waits in front.
x² gives 2x.
So 2xy.
yz³ has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = 2xy.
Careful!
A fixed letter is not 0!
In x²y, y stays in front.
Only a term with no x at all gives 0.
2 Partial by y
For ∂f/∂y, only y changes.
x and z stay fixed: they act like plain numbers.
They are purple now.
Term by term.
In x²y, x² is fixed and waits in front.
y gives 1.
So x².
In yz³, z³ is fixed and waits in front.
y gives 1.
So z³.
So ∂f/∂y = x² + z³.
3 Partial by z
For ∂f/∂z, only z changes.
x and y stay fixed: they act like plain numbers.
They are purple now.
Term by term.
x²y has no z: it is a fixed number, so it gives 0.
In yz³, y is fixed and waits in front.
z³ gives 3z².
So 3yz².
Drop the zeros: ∂f/∂z = 3yz².
4 Check
The slope between two very close points.
Move x by 0.001: the change in f, over 0.001, is about 4.
And ∂f/∂x at that point is 4.
Almost the same.
Right!
Now move y by 0.001: the change in f, over 0.001, is about 2.
And ∂f/∂y at that point is 2.
Almost the same.
Right!
Now move z by 0.001: the change in f, over 0.001, is about 6.01.
And ∂f/∂z at that point is 6.
Almost the same.
Right!
Let's look back.
First: partial by x.
Then: partial by y.
Then: partial by z.
Last: check.