Find the directional derivative fa(1,2) for the function f with u⃗=(3 i⃗-4 j⃗) / 5 f(x, y)=x^2-y

step 1 What are given question, we need to find the diasland derivative of this function at this point

Find the directional derivative fa(1,2) for the function f...

Key Concepts

Directional Derivative

The directional derivative represents the rate at which a function changes at a given point in a specified direction. It generalizes the notion of partial derivatives by considering arbitrary directions instead of just the coordinate axes, and it is obtained by projecting the gradient of the function onto a unit vector in the desired direction.

Gradient Vector

The gradient vector of a function is a vector comprised of its partial derivatives, and it points in the direction of the greatest rate of increase of the function. The magnitude of the gradient gives the rate of increase in that direction, and it is fundamental in calculating the directional derivative by using the dot product with a unit vector.

Unit Vector

A unit vector is a vector of length one that specifies direction without regard to magnitude. In the context of directional derivatives, using a unit vector ensures that the derivative measures the rate of change in that precise direction, without scaling from the vector’s magnitude.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (typically coordinate vectors) and returns a single number. It is used to project one vector onto another; when calculating a directional derivative, the dot product of the gradient vector and a direction's unit vector gives the desired rate of change in that direction.

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