Step by step
The question
Question: find the partial derivatives of f(x, y) = x⁴ − 3xy² + 6y.
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.
x⁴: the power 4 comes down.
The power goes down by 1.
So 4x³.
In 3xy², 3y² is fixed and waits in front.
x gives 1.
So 3y².
6y has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = 4x³ − 3y².
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.
x⁴ has no y: it is a fixed number, so it gives 0.
In 3xy², 3x is fixed and waits in front.
y² gives 2y.
So 6xy.
6y gives 6.
Drop the zeros: ∂f/∂y = −6xy + 6.
3 Check
Let's check.
Here is the surface z = f(x, y) near (1, 2).
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 −7.99.
And ∂f/∂x at that point is −8.
Almost the same.
Right!
Now move y by 0.001: the change in f, over 0.001, is about −6.
And ∂f/∂y at that point is −6.
Almost the same.
Right!
Let's look back.
First: partial by x.
Then: partial by y.
Last: check.