Let f be a function of two variables that has continuous...

Let f be a function of two variables that has continuous partial derivatives and consider the points $A(1,3), B(3,3),$$C(1,7),$ and $D(6,15) .$$ The directional derivative of $f$ at $A$ in the direction of the vector $A B$ is 3 and the directional derivative at $A$ in the direction of $\over line{A C}$ is $26 .$ Find the directional derivative of $f$ at $A$ in the direction of the vector $\overline{A D}$

Key Concepts

Directional Derivative

The directional derivative of a function at a point in a specific direction is a measure of how fast the function is changing at that point as one moves in that direction. It is computed as the dot product of the gradient vector of the function at that point and a unit vector in the given direction.

Gradient Vector

The gradient vector of a function of several variables is a vector that contains all of its first-order partial derivatives. This vector points in the direction of the greatest rate of increase of the function and its magnitude gives the rate of increase in that direction.

Unit Vector

A unit vector is a vector with a magnitude of one and is used to specify direction without any scaling. In the context of directional derivatives, the given direction vector must be converted to a unit vector to ensure that the rate of change is measured per unit length.

Dot Product

The dot product is a fundamental vector operation that multiplies corresponding components of two vectors and sums them to produce a scalar. In calculating the directional derivative, the dot product of the gradient vector and the normalized direction vector yields the rate of change of the function in that direction.

Systems of Equations

When multiple directional derivatives in different directions are provided, they can be used to set up a system of equations that relates the known directional derivatives to the unknown partial derivatives of the function. Solving this system allows one to determine the specific values of these partial derivatives, which can then be used to compute the directional derivative in any desired direction.

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