an 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,x) and er(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

a. To find , we need to take the partial derivatives of with respect to each variable. Now, we can evaluate these partial derivatives at the point : So, at is .b. The directional derivative of at in the direction of is given by the off P u u u = \langle 1,-1,\sqrt {2}\rangle |u| = \sqrt{1^2 + (-1)^2 + (\sqrt{2})^2} = \sqrt{ = 2 u_{\text{unit}} = \frac{u}{|u|} = \frac{1}{2}\langle 1,-1,\sqrt {2}\rangle = \langle \frac{1}{2},-\frac{1}{2},\frac{\sqrt{2}}{2}\rangle \nabla f(P) = \langle 4 + 8m, -2, 0 \rangle \frac{\partial f}{\partial u} = \nabla f(P) \cdot u_{\text{unit}} = (4 + 8m) \cdot \frac{1}{2} + (-2) \cdot \left(-\frac{1}{2}\right) + 0 \cdot \frac{\sqrt{2}}{2} = 2 + 4m + 1 = 3 + 4m f(x,y,z) P(2,1,-1) u=\langle 1,-1,\sqrt {2}\rangle 3 + 4m$.