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 $\overrightarrow{A B}$ is 3 , and the directional derivative at $A$ in the direction of $\overrightarrow{A C}$ is 26 . Find the directional derivative of $f$ at $A$ in the direction of the vector $\overrightarrow{A D}$.

Key Concepts

Directional Derivative

The directional derivative is a measure of how a function changes as one moves in a specified direction from a given point. It generalizes the concept of the derivative by taking into account the direction in which the change is being considered, rather than just along the coordinate axes.

Dot Product

The dot product is an algebraic operation that multiplies two vectors, resulting in a scalar. In the context of directional derivatives, the dot product between the gradient vector and a unit vector in the desired direction yields the rate of change of the function in that direction.

Unit Vector

A unit vector is a vector of length one that indicates direction. When computing directional derivatives, the vector representing the direction must be normalized to a unit vector so that the result reflects the true rate of change per unit distance in that specific direction.

Continuous Partial Derivatives

Continuous partial derivatives signify that the function's rates of change with respect to each variable do not have abrupt jumps or discontinuities. This continuity is a key condition for applying many differential calculus techniques, including the use of the gradient and directional derivatives.

Gradient Vector

The gradient vector is an array of partial derivatives that points in the direction of the greatest rate of increase of a function. Its components are the partial derivatives with respect to each variable, and it plays a crucial role in computing directional derivatives via the dot product.

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