Directional Derivative

The magnitude of the directional derivative of the function [Tex]f\left( {x,y} \right) = {x^2} + 3{y^2}[/Tex] in a direction normal to the circle [Tex]\,{x^2} + {y^2} = 2,[/Tex] at the point [Tex](1,1),[/Tex] is

The directional derivative of the function [Tex]f(x, y, z) = x + y[/Tex] at the point [Tex]P(1,1,0)[/Tex] along the direction [Tex]\overrightarrow i + \overrightarrow j [/Tex] is

For a scalar function [Tex]\,f\left( {x,y,z} \right) = {x^2} + 3{y^2} + 2{z^2},\,\,[/Tex] the directional derivative at the point [Tex]P(1,2,-1)[/Tex] in the direction of a vector [Tex]\widehat i - \widehat j + 2\widehat k\,\,[/Tex] is

The directional derivative of [Tex]\,\,f\left( {x,y,z} \right) = 2{x^2} + 3{y^2} + {z^2}\,\,[/Tex] at the point [Tex]P(2,1,3)[/Tex] in the direction of the vector [Tex]{\mkern 1mu} \vec a = \widehat i - 2\widehat k{\mkern 1mu} [/Tex] is _________.

The directional derivative of the following function at [Tex](1, 2)[/Tex] in the direction of [Tex](4i+3j)[/Tex] is : [Tex]f\left( {x,y} \right) = {x^2} + {y^2}[/Tex]

Consider a vector[Tex]\vec{u} = 2\hat{x} + \hat{y} + 2\hat{z}[/Tex], where [Tex]\hat{x}, \hat{y}, \hat{z} [/Tex]represent unit vectors along the coordinate axes x,y z respectively. The directional derivative of the function [Tex]f(x, y, z) = 2 \ln(xy) + \ln(yz) + 3 \ln(xz)[/Tex] at the point (x, y, z) = (1, 1, 1) in the direction of [Tex]\vec{u}[/Tex] is