Question

Calculate all four second-order partial derivatives of X"(x,y)=4x²y 3xy³?

Answer 1

The student's question involves calculating the second-order partial derivatives of the function X"(x,y)=4x²y + 3xy³. By taking repeated derivatives, one can find X"_xx, X"_yy, X"_xy, and X"_yx, which are the partial derivatives with respect to x and y individually and combined.

Step-by-step explanation:

The student has asked to calculate all four second-order partial derivatives of the function X"(x,y)=4x²y + 3xy³. The second-order partial derivatives involve taking the derivative twice, first with respect to one variable while holding the other constant, and then the second derivative with respect to the same or the other variable.

Let's calculate the four second-order partial derivatives step by step:

1. First, take the partial derivative of X" with respect to x, which will give us X"_x.
2. Then take the derivative of X"_x with respect to x again, resulting in X"_xx.
3. Similarly, the partial derivative of X" with respect to y gives us X"_y.
4. Then take the derivative of X"_y with respect to y again to get X"_yy.
5. For mixed derivatives, first take the derivative of X"_x with respect to y to obtain X"_xy.
6. Alternatively, take the derivative of X"_y with respect to x to find X"_yx, noting that X"_xy should equal to X"_yx by Clairaut's theorem if the function is sufficiently smooth.

After performing these differentiations, we obtain the values for X"_xx, X"_yy, X"_xy, and X"_yx, which are the four second-order partial derivatives sought after.

Answer 2

The second-order partial derivatives of the function X"(x,y) = 4x²y + 3xy³ include ∂²X"/∂x² = 8y and ∂²X"/∂y² = 18xy. The mixed second-order partial derivatives are both ∂²X"/∂y∂x = ∂²X"/∂x∂y = 8x + 9y².

Step-by-step explanation:

The question asks to calculate all four second-order partial derivatives of the function X"(x,y) = 4x²y + 3xy³. To find these, we follow the steps below:

1. First, take the partial derivative of X" with respect to x to get a first-order partial derivative with respect to x, which is 8xy + 3y³.
2. Then, differentiate the above result again with respect to x to get the second-order partial derivative with respect to x, ∂²X"/∂x² which is 8y.
3. Next, take the partial derivative of X" with respect to y to get a first-order partial derivative with respect to y, which is 4x² + 9xy².
4. Finally, differentiate the above result again with respect to y to get the second-order partial derivative with respect to y, ∂²X"/∂y² which is 18xy.

For the mixed second-order partial derivatives, we also need to:

1. Take the partial derivative of the first-order partial derivative with respect to x (obtained in step 1) with respect to y, which gives ∂²X"/∂y∂x = 8x + 9y².
2. Take the partial derivative of the first-order partial derivative with respect to y (obtained in step 3) with respect to x, which gives ∂²X"/∂x∂y = 8x + 9y².

The mixed partial derivatives are equal, which is consistent with Clairaut's theorem on the equality of mixed partial derivatives assuming the function is nice enough (in this case, continuous and differentiable).