Question

Find the directional derivative of f (x, y,z) = x2y3 − 4xz in the direction of f (x, y,z) = x2y3 − 4xz

Answer 1

$$f(x, y, z)=x^2y^3-4xz$$

$$\vec v=\langle-1,2,0\rangle, Point(0,0,1)$$

$$\nabla f=\dfrac{\partial f}{\partial x}\vec i +\dfrac{\partial f}{\partial y}\vec j +\dfrac{\partial f}{\partial z}\vec k$$

$$=(2xy^3-4z)\vec i +(3x^2y^2)\vec j -4x\vec k$$

$$\nabla f|_{(0,0,1)}=-4\vec i$$

$$|\vec v|=\sqrt{(-1)^2+(2)^2+(0)^2}=\sqrt{5}$$

$$D_{\vec v}=\nabla f|_{(0,0,1)}\cdot\dfrac{\vec v}{|\vec v|}=-4\vec i\cdot\dfrac{-\vec i +2\vec j +0\vec k }{\sqrt{5}}=\dfrac{4\sqrt{5} }{5}$$