Question

Find all the second-order partial derivatives of the following function.
w =2x sin (4x^2y)

Answer 1

Given: w = 2x sin(4x^2y),

First, we take the partial derivative with respect to x:
∂w/∂x = 4x^2y cos(4x^2y) + 2sin(4x^2y)

Next, we take the partial derivative of this result with respect to x again:

∂^2w/∂x^2 = (8xy cos(4x^2y) - 32x^4y^2 sin(4x^2y))

Now, we take the partial derivative with respect to y:

∂w/∂y = 2x(4x^2 cos(4x^2y))

And, we take the partial derivative of this result with respect to y again:

∂^2w/∂y^2 = -32x^4 sin(4x^2y)

Therefore, the second-order partial derivatives of the function w = 2x sin(4x^2y) are:

∂^2w/∂x^2 = (8xy cos(4x^2y) - 32x^4y^2 sin(4x^2y))
∂^2w/∂y^2 = -32x^4 sin(4x^2y)