Word Count : 507

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The directional derivative of a vector-valued function (f : \mathbb{R}^4 \to \mathbb{R}^4) at a point ((x_0, y_0, z_0, w_0)) in the direction of a vector (v) is given by: [ D_v f(x_0, y_0, z_0, w_0) = J_f(x_0, y_0, z_0, w_0) \cdot \frac{v}{|v|} ] where (J_f) is the Jacobian matrix of (f) and (\frac{v}{|v|}) is the unit vector in the direction of (v). --- Step 1: Compute the Jacobian of (f(x, y, z, w) = (x^2 y, xyz, x^2 + y^2, zw^2)). [ J_f = \begin{bmatrix} \frac{\partial}{\partial x}(x^2 y) & \frac{\partial}{\partial y}(x^2 y) & \frac{\partial}{\partial z}(x^2 y) & \frac{\partial}{\partial w}(x^2 y) \ \frac{\partial}{\partial x}(xyz) & \frac{\partial}{\partial y}(xyz) & \frac{\partial}{\partial z}(xyz) & \frac{\partial}{\partial w}(xyz) \ \frac{\partial}{\partial x}(x^2 + y^2) & \frac{\partial}{\partial y}(x^2 + y^2) & \frac{\partial}{\partial z}(x^2 + _________ _________ ______ _________ ________ ______ ________.
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