Step by step
The question
Question: find the partial derivatives of f(x, y) = x²y² − 4xy at (2, 1).
1 Partial by x
The partial derivative ∂f/∂x is the slope when only x changes.
For ∂f/∂x, only x changes.
y stays fixed: it acts like a plain number.
It is purple now.
Term by term.
In x²y², y² is fixed and waits in front.
x² gives 2x.
So 2xy².
In 4xy, 4y is fixed and waits in front.
x gives 1.
So 4y.
So ∂f/∂x = 2xy² − 4y.
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 stays fixed: it acts like a plain number.
It is purple now.
Term by term.
In x²y², x² is fixed and waits in front.
y² gives 2y.
So 2x²y.
In 4xy, 4x is fixed and waits in front.
y gives 1.
So 4x.
So ∂f/∂y = 2x²y − 4x.
3 Put the point in
Put x = 2, y = 1 into ∂f/∂x.
Work out each term.
So f_x(2, 1) = 0.
Put x = 2, y = 1 into ∂f/∂y.
Work out each term.
So f_y(2, 1) = 0.
4 Check
Let's check.
Here is the surface z = f(x, y) near (2, 1).
On the orange curve only x changes.
∂f/∂x is its slope.
On the purple curve only y changes.
∂f/∂y is its slope.
The slope between two very close points.
Move x by 0.001: the change in f, over 0.001, is about 0.
And ∂f/∂x at that point is 0.
Almost the same.
Right!
Now move y by 0.001: the change in f, over 0.001, is about 0.
And ∂f/∂y at that point is 0.
Almost the same.
Right!
Let's look back.
First: partial by x.
Then: partial by y.
Then: put the point in.
Last: check.