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

step 1 We have D u squared f of x y. This is equal to D u into D u f of x comma y. Now we have x comma

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

The second directional derivative of $f(x, y)$ is

$$D_{\mathfrak{u}}^{2} f(x, y)=D_{\mathrm{u}}\left[D_{\mathrm{u}} f(x, y)\right]$$

$$f(x, y)=x^{3}+5 x^{2} y+y^{3}$ and $\mathbf{u}=\left[\frac{3}{5}, \frac{4}{5}\right],calculate $D_{\mathrm{n}}^{2} f(2,1)$$

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Key Concepts

Directional Derivative

The directional derivative measures the rate at which a function changes at a specific point in the direction of a given unit vector. It is calculated using the dot product of the function's gradient and the unit vector, providing insight into the function's behavior along that particular direction.

Second Directional Derivative

The second directional derivative evaluates how the first directional derivative changes along the same direction. It provides information on the curvature of the function in that direction and is typically computed using the Hessian matrix, resulting in a quadratic form involving the unit vector.

Hessian Matrix

The Hessian matrix is a square matrix that contains all the second partial derivatives of a function. It encapsulates the local curvature properties of the function and is a key tool in analyzing its second-order behavior. When the Hessian is combined with a direction vector, it allows for the computation of the second directional derivative.

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