Step by step
The question
Question: find the second partial derivatives of f(x, y) = x⁴y² − 3y.
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 x⁴y², y² is fixed and waits in front.
x⁴ gives 4x³.
So 4x³y².
3y has no x: it is a fixed number, so it gives 0.
Drop the zeros: ∂f/∂x = 4x³y².
2 Partial by y
Term by term.
In x⁴y², x⁴ is fixed and waits in front.
y² gives 2y.
So 2x⁴y.
3y gives 3.
So ∂f/∂y = 2x⁴y − 3.
3 The second partials
Now take each answer's partial derivatives again.
f_xy means: first by x, then by y.
Term by term.
In 4x³y², 4y² is fixed and waits in front.
x³ gives 3x².
So 12x²y².
So f_xx = 12x²y².
Term by term.
In 4x³y², 4x³ is fixed and waits in front.
y² gives 2y.
So 8x³y.
So f_xy = 8x³y.
Term by term.
In 2x⁴y, 2x⁴ is fixed and waits in front.
y gives 1.
So 2x⁴.
3 has no y: it is a fixed number, so it gives 0.
Drop the zeros: f_yy = 2x⁴.
Term by term.
In 2x⁴y, 2y is fixed and waits in front.
x⁴ gives 4x³.
So 8x³y.
3 has no x: it is a fixed number, so it gives 0.
Drop the zeros: f_yx = 8x³y.
4 Check
Let's check.
The two mixed ones, f_xy and f_yx, must come out the same.
They do.
Right!
Let's look back.
First: partial by x.
Then: partial by y.
Then: the second partials.
Last: check.