Step by step
The question
Question: find the partial derivatives of f(x, y) = 5x³ − 2x²y + y.
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.
5x³: the power 3 comes down: 5 times 3 is 15.
The power goes down by 1.
So 15x².
In 2x²y, 2y is fixed and waits in front.
x² gives 2x.
So 4xy.
y has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = 15x² − 4xy.
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.
5x³ has no y: it is a fixed number, so it gives 0.
In 2x²y, 2x² is fixed and waits in front.
y gives 1.
So 2x².
y gives 1.
Drop the zeros: ∂f/∂y = −2x² + 1.
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 7.01.
And ∂f/∂x at that point is 7.
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!
Let's look back.
First: partial by x.
Then: partial by y.
Last: check.