Step by step
The question
Question: find the partial derivatives of f(x, y, z) = xyz + z².
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 xyz, yz is fixed and waits in front.
x gives 1.
So yz.
z² has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = yz.
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 xyz, xz is fixed and waits in front.
y gives 1.
So xz.
z² has no y: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂y = xz.
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.
In xyz, xy is fixed and waits in front.
z gives 1.
So xy.
z²: the power 2 comes down.
The power goes down by 1.
So 2z.
So ∂f/∂z = xy + 2z.
4 Check
The slope between two very close points.
Move x by 0.001: the change in f, over 0.001, is about 2.
And ∂f/∂x at that point is 2.
Almost the same.
Right!
Now move y by 0.001: the change in f, over 0.001, is about 1.
And ∂f/∂y at that point is 1.
Almost the same.
Right!
Now move z by 0.001: the change in f, over 0.001, is about 4.
And ∂f/∂z at that point is 4.
Almost the same.
Right!
Let's look back.
First: partial by x.
Then: partial by y.
Then: partial by z.
Last: check.