Find the directional derivative of f at P in the direction of 𝐚 . f(x, y)=4 x^3 y^2 ; P(2,1) ; 𝐚

step 1 In this question the vector A is given here we need to find the unit vector u which is nothing b

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

Key Concepts

Directional Derivative

The directional derivative of a function at a point in a given direction measures the rate at which the function changes at that point along the specified direction. It is defined as the dot product of the gradient of the function at that point and a unit vector in the given direction, which captures both the magnitude and the orientation of the rate of change.

Gradient

The gradient of a multivariable function is a vector comprising its partial derivatives with respect to each variable. It points in the direction of the steepest ascent of the function and its magnitude indicates the rate of increase in that direction. Calculating the gradient is a key step in finding the directional derivative.

Unit Vector

A unit vector is a vector with a magnitude of one and is used to specify direction without scaling the rate of change. When determining the directional derivative, the provided direction vector must be normalized (converted into a unit vector) to ensure that the derivative reflects the rate of change per unit length in that direction.

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, the dot product of the gradient and the unit vector in the specified direction yields the component of the gradient in that direction, which is the directional derivative.

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