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To solve this problem, we need to find the directional derivative of the function \( f(x, y, z, w) = (x^2 y, xyz, x^2 + y^2, zw^2) \) at the point \( (1, 2, -1, -2) \) in the direction of the vector \( v = (1, 0, -2, 2) \). The directional derivative of a vector-valued function \( f \) at a point \( (x_0, y_0, z_0, w_0) \) in the direction of a vector \( v = (v_1, v_2, v_3, v_4) \) is given by the dot product of the gradient of \( f \) and the unit vector in the direction of \( v \): \[ D_v f = \nabla f \cdot \frac{v}{\|v\|} \] ### Step 1: Compute the gradient \( \nabla f \) The gradient \( \nabla f \) is the Jacobian matrix of partial derivatives of each component ____ ______ ______ ____ __________ ________ _____ ____.
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