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Homedirectional derivative

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# directional derivative

Let $U$ be an open set in $\mathbbmss{R}^{n}$ and $f\colon U\to\mathbbmss{C}$ is a differentiable
function. If $u\in U$ and $v\in\mathbb{R}^{n}$, then the
*directional derivative* of $f$ in the direction of $v$ is

$(D_{v}f)(u)=\frac{d}{ds}f(u+sv)\Big|_{{s=0}}.$ |

In other words, $(D_{v}f)(u)$ measures how $f$ changes in the direction of $v$ from $u$.

Alternatively,

$\displaystyle(D_{v}f)(u)$ | $\displaystyle=$ | $\displaystyle\lim_{{h\to 0}}\frac{f(u+hv)-f(u)}{h}$ | ||

$\displaystyle=$ | $\displaystyle Df(u)\cdot v,$ |

where $Df$ is the Jacobian matrix of $f$.

# Properties

Let $u\in U$.

1. $D_{v}f$ is linear in $v$. If $v,w\in\mathbbmss{R}^{n}$ and $\lambda,\mu\in\mathbbmss{R}$, then

$D_{{\lambda v+\mu w}}f(u)=\lambda D_{{v}}f(u)+\mu D_{{w}}f(u).$ In particular, $D_{0}f=0$.

2. If $f$ is twice differentiable and $v,w\in\mathbbmss{R}^{n}$, then

$\displaystyle D_{v}D_{w}f(u)$ $\displaystyle=$ $\displaystyle\frac{\partial^{2}}{\partial s\partial t}f(u+sv+tw)\Big|_{{s=0}},$ $\displaystyle=$ $\displaystyle v^{T}\cdot\operatorname{Hess}f(u)\cdot w,$ where $\operatorname{Hess}$ is the Hessian matrix of $f$.

# Example

For example, if $f\left(\begin{array}[]{c}x\\ y\\ z\end{array}\right)=x^{2}+3y^{2}z$, and we wanted to find the derivative at the point $\mathbf{a}=\left(\begin{array}[]{c}1\\ 2\\ 3\end{array}\right)$ in the direction $\vec{v}=\left[\begin{array}[]{c}1\\ 1\\ 1\end{array}\right]$, our equation would be

$\displaystyle\lim_{{h\rightarrow 0}}\frac{1}{h}\left((1+h)^{2}+3(2+h)^{2}(3+h)% -37\right)$ | $\displaystyle=$ | $\displaystyle\lim_{{h\rightarrow 0}}\frac{1}{h}(3h^{3}+37h^{2}+50h)$ | ||

$\displaystyle=$ | $\displaystyle\lim_{{h\rightarrow 0}}3h^{2}+37h+50=50$ |

## Mathematics Subject Classification

26B12*no label found*26B10

*no label found*

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