Second partial derivatives of f(x, y) = x^3 y^2

Step by step

The question

Question: find the second partial derivatives of f(x, y) = x³y².

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 x³y², y² is fixed and waits in front.

x³ gives 3x².

So 3x²y².

So ∂f/∂x = 3x²y².

2 Partial by y

Term by term.

In x³y², x³ is fixed and waits in front.

y² gives 2y.

So 2x³y.

So ∂f/∂y = 2x³y.

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 3x²y², 3y² is fixed and waits in front.

x² gives 2x.

So 6xy².

So f_xx = 6xy².

Term by term.

In 3x²y², 3x² is fixed and waits in front.

y² gives 2y.

So 6x²y.

So f_xy = 6x²y.

Term by term.

In 2x³y, 2x³ is fixed and waits in front.

y gives 1.

So 2x³.

So f_yy = 2x³.

Term by term.

In 2x³y, 2y is fixed and waits in front.

x³ gives 3x².

So 6x²y.

So f_yx = 6x²y.

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.