The second directional derivative of f(x, y) is Da^2 f(x, y)=Da[Da f(x, y)] If f(x, y)=x^3+5 x^2

step 1 Today we are going to solve for number 39. Here function in x comma y is x cube plus 5 x square

The second directional derivative of f(x, y) is Da^2 f(x,...

The second directional derivative of $f(x, y)$ is $$D_{\mathrm{a}}^{2} f(x, y)=D_{\mathrm{a}}\left[D_{\mathrm{a}} f(x, y)\right]$$
If $f(x, y)=x^{3}+5 x^{2} y+y^{3}$ and $\mathbf{u}=\left\langle\frac{3}{5}, \frac{4}{5}\right\rangle,$$ calculate $ D_{\mathbf{u}}^{2} f(2,1)$$

Key Concepts

Partial Derivatives

Partial derivatives represent the derivatives of a multivariable function with respect to one variable while keeping the other variables constant. They are fundamental in constructing the gradient and Hessian matrices, serving as building blocks for understanding and computing how a function changes in different directions within its domain.

Hessian Matrix

The Hessian matrix is a square matrix composed of all the second-order partial derivatives of a multivariable function. This matrix plays a key role in evaluating second derivatives, including the second directional derivative, as it encapsulates information about the local curvature and can be used to determine the nature of critical points in optimization problems.

Directional Derivative

A directional derivative is a measure of how a function changes as one moves in a specific direction from a given point. It generalizes the concept of the derivative by extending it from one-dimensional to multi-dimensional functions, using the gradient vector and projecting it onto a unit vector that specifies the direction of interest.

Second Directional Derivative

The second directional derivative quantifies the rate at which the directional derivative itself changes as one moves in a specified direction. It is essentially the derivative of the directional derivative, providing insight into the curvature of the function along that direction, and is particularly useful in analyzing the function's concavity or convexity.

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