Find all the second-order partial derivatives of the following function: f(x, y) = x2y3

We will find the second-order partial derivatives of the function f.

Answer: The second-order partial derivatives are ∂2f / ∂x2 = 2y3, ∂2f / ∂y2 = 6x2y,  ∂2f / ∂x∂y = 6xy2 and ∂2f / ∂y∂x = 6xy2.

We will find ∂2f / ∂x2, ∂2f / ∂y2, ∂2f / ∂x∂y and ∂2f / ∂y∂x.

Explanation: 

The given function is f(x, y) = x2y3.

∂f / ∂x = 2xy3

∂2f / ∂x2 = ∂(2xy3) / ∂x = 2y3

∂2f / ∂x∂y = ∂(2xy3) / ∂y = 6xy2

∂f / ∂y = 3x2y2

∂2f / ∂y2 = ∂(3x2y2) / ∂y = 6x2y

∂2f / ∂y∂x = ∂(3x2y2) / ∂x = 6xy2

You can use Cuemath's Partial Derivative Calculator that will help you find the first-order partial derivatives.

Thus, the second-order partial derivatives are ∂2f / ∂x2 = 2y3, ∂2f / ∂y2 = 6x2y,  ∂2f / ∂x∂y = 6xy2 and ∂2f / ∂y∂x = 6xy2.

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