Questions: Let f(x, y) = 4x^2y + xy^4. Find the second partial derivative of f with respect to x, the second partial derivative of f with respect to y, the mixed partial derivative of f first with respect to x then y, and the mixed partial derivative of f first with respect to y then x.

Solution Steps

To find the second partial derivatives of the function \( f(x, y) = 4x^2y + xy^4 \), we will first compute the first partial derivatives with respect to \( x \) and \( y \). Then, we will differentiate these first partial derivatives again with respect to \( x \) and \( y \) to obtain the second partial derivatives: \(\frac{\partial^2 f}{\partial x^2}\), \(\frac{\partial^2 f}{\partial y^2}\), \(\frac{\partial^2 f}{\partial x \partial y}\), and \(\frac{\partial^2 f}{\partial y \partial x}\).

To find \( \frac{\partial^2 f}{\partial x^2} \), we differentiate \( f(x, y) = 4x^2y + xy^4 \) twice with respect to \( x \): \[ \frac{\partial^2 f}{\partial x^2} = 8y \]

Next, we compute \( \frac{\partial^2 f}{\partial y^2} \) by differentiating \( f(x, y) \) twice with respect to \( y \): \[ \frac{\partial^2 f}{\partial y^2} = 12xy^2 \]

We will now find the mixed partial derivatives \( \frac{\partial^2 f}{\partial x \partial y} \) and \( \frac{\partial^2 f}{\partial y \partial x} \). Both derivatives are computed as follows: \[ \frac{\partial^2 f}{\partial x \partial y} = 4(2x + y^3) \] \[ \frac{\partial^2 f}{\partial y \partial x} = 4(2x + y^3) \]

Final Answer