Q2. Given f(x,y,z)=x^2-y^2+2mx^2 and a point P(2,1,-1) then a. Find forall f(x,y,z) and (r(2.1,-1) b. Find the directional derivative of f(x,y,z) at P(2,1,-1) in the direction of u=langle 1,-1,sqrt (2)rangle

To find the gradient of the function \( f(x, y, z) = x^2 - y^2 + 2mx^2 \) at the point \( P(2, 1, -1) \), we need to compute the partial derivatives of with respect to , , and .The partial derivatives are: Evaluating these partial derivatives at the point \( P(2, 1, -1) \): So, the gradient vector at \( P(2, 1, -1) \) is: Next, we need to find the directional derivative of at \( P(2, 1, -1) \) in the direction of the unit vector .First, we normalize the unit vector : The directional derivative of in the direction of is given by the dot product of the gradient vector and the unit vector : So, the directional derivative of at \( P(2, 1, -1) \) in the direction of is .