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 \( f \) with respect to \( x \), \( y \), and \( z \).<br /><br />The partial derivatives are:<br />\[ \frac{\partial f}{\partial x} = 2x + 4mx \]<br />\[ \frac{\partial f}{\partial y} = -2y \]<br />\[ \frac{\partial f}{\partial z} = 0 \]<br /><br />Evaluating these partial derivatives at the point \( P(2, 1, -1) \):<br />\[ \left. \frac{\partial f}{\partial x} \right|_{(2, 1, -1)} = 2(2) + 4m(2) = 4 + 8m \]<br />\[ \left. \frac{\partial f}{\partial y} \right|_{(2, 1, -1)} = -2(1) = -2 \]<br />\[ \left. \frac{\partial f}{\partial z} \right|_{(2, 1, -1)} = 0 \]<br /><br />So, the gradient vector at \( P(2, 1, -1) \) is:<br />\[ \nabla f(2, 1, -1) = \langle 4 + 8m, -2, 0 \rangle \]<br /><br />Next, we need to find the directional derivative of \( f \) at \( P(2, 1, -1) \) in the direction of the unit vector \( u = \langle 1, -1, \sqrt{2} \rangle \).<br /><br />First, we normalize the unit vector \( u \):<br />\[ \|u\| = \sqrt{1^2 + (-1)^2 + (\sqrt{2})^2} = \sqrt{1 + 1 + 2} = \sqrt{4} = 2 \]<br />\[ \hat{u} = \left\langle \frac{1}{2}, \frac{-1}{2}, \frac{\sqrt{2}}{2} \right\rangle \]<br /><br />The directional derivative of \( f \) in the direction of \( u \) is given by the dot product of the gradient vector and the unit vector \( \hat{u} \):<br />\[ D_{\hat{u}} f(2, 1, -1) = \nabla f(2, 1, -1) \cdot \hat{u} \]<br />\[ = \langle 4 + 8m, -2, 0 \rangle \cdot \left\langle \frac{1}{2}, \frac{-1}{2}, \frac{\sqrt{2}}{2} \right\rangle \]<br />\[ = \left( 4 + 8m \right) \cdot \frac{1}{2} + (-2) \cdot \frac{-1}{2} + 0 \cdot \frac{\sqrt{2}}{2} \]<br />\[ = 2 + 4m + 1 \]<br />\[ = 3 + 4m \]<br /><br />So, the directional derivative of \( f \) at \( P(2, 1, -1) \) in the direction of \( u \) is \( 3 + 4m \).