Find the directional derivative of f at P in the direction of 𝐚 . f(x, y)=y^2 lnx ; P(1,4) ; 𝐚=-

step 1 In this question the given vector a is equal to minus 3i plus 3 j we need the unit vector u whic

Find the directional derivative of f at P in the direction...

Key Concepts

Directional Derivative

The directional derivative of a function at a given point measures the rate at which the function changes as one moves from that point in a specified direction. It is calculated by taking the dot product of the gradient vector at that point with a unit vector that represents the given direction. This concept is central to understanding how multivariable functions behave along arbitrary directions, not just along the coordinate axes.

Gradient

The gradient of a function of several variables is a vector that contains all of its partial derivatives. It points in the direction of the steepest ascent of the function and its magnitude indicates the rate of increase in that direction. The gradient is fundamental in the computation of directional derivatives and in optimization problems.

Partial Derivatives

Partial derivatives are the derivatives of a multivariable function with respect to one variable while keeping other variables constant. They measure how the function changes as each independent variable is varied. These are used to form the gradient vector, which encapsulates all the directional rates of change of the function.

Unit Vector

A unit vector has a magnitude of one and is used to indicate direction without affecting the scale. When calculating the directional derivative, the given direction vector must be normalized to produce a unit vector, ensuring that the computed derivative accurately reflects only the directional rate of change.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In the context of directional derivatives, it is used to combine the information of the gradient vector with the direction vector, effectively projecting the gradient onto the direction of interest.

Transcript

Please give Ace some feedback