정확한 표기	의미
Derivative	derivative of $f$ with respect to $x$	the slope of $f$ at $x$
Partial derivative	partial derivative of $\mathsf{f}=[f_1,\dots,f_m]$, $\frac{\partial f_i}{\partial x_j}$	the slope of $f_i$ at $x_j$
Gradient	gradient of $f$ at $\bold{x}$	a vector(or matrix, if $f$ is multivariate) whose components are the partial derivatives of $f$ at $\bold{x}$
Directional derivative	directional derivative of $f$ with respect to $\bold{x}$, in the direction of $\bold{u}$	the slope of $f$ at $\bold{x}$ in the direction of $\bold{u}$

• $f(\bold{x}) = 2x^2+5xy+3y^3+yz+z^2$, where $\bold{x}=(x,y,z)$을 예시로 위의 개념을 계산해보면,
  -이 경우, $f:\reals^3\to\reals$ 임을 주의하라!

	Dim	계산 결과
Derivative	$\reals\to\reals$	$f_x'=4x+5y$, $f_y'=5x+9y^2+z$, $f_z'=y+2z$
Partial derivative	$\reals^3\to\reals$	$\frac{\partial f_x}{\partial x}=4x+5y$, $\frac{\partial f_y}{\partial y}=5x+9y^2+z$, $\frac{\partial f_z}{\partial z}=y+2z$
Gradient	$\reals^3\to\reals^3$	$[\frac{\partial f_x}{\partial x},\frac{\partial f_y}{\partial y},\frac{\partial f_z}{\partial z}]=[4x+5y, 5x+9y^2+z, y+2z]$
Directional derivative	$\reals^3\to\reals$	in the direction of $[0, 1, 0]=\\(\text{gradinet of }f)^\top[0,1,0]=5x+9y^2+z$

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Derivative

-$f:\reals^ \to \reals$
-derivative of $f$ with respect to $x$

$\phi(x)=\lim_{t\to x}\frac{f(t)-f(x)}{t-x}$

provided this limit exists.

Partial derivative

-$\mathsf{f}:$ an open set $E \subset \reals^n \to \reals^m$
-$\{\bold{e}_1,\bold{e}_2,\dots,\bold{e}_n\}$ : the standard basis of $\reals^n$
-$\{\bold{u}_1,\bold{u}_2,\dots,\bold{u}_m\}$ : the standard basis of $\reals^m$
-the component of $\mathsf{f}$ are the real functions $f_1, f_2, \dots, f_m$ where $f_i:\reals^n \to \reals$

$(D_jf_i)(\bold{x}) = \lim_{t\to 0}\frac{f_i(\bold{x}+te_j) - f_i(\bold{x})}{t}$

-Writing $f_i(x_1,\dots,x_n)$ in place of $f_i(\bold{x})$, we can see that $D_jf_i$ is the derivative of $f_i$ with respect to $x_j$, keeping the other variables fixed.
-$D_jf_i$ is called a partial derivative.

-The other notation is

$\frac{\partial f_i}{\partial x_j}$

Gradient

-$f:\reals^n\to\reals$
-$\{\bold{e}_1,\bold{e}_2,\dots,\bold{e}_n\}$ : standard basis of $\reals^n$
-gradient of $f$ at $\bold{x}$

$(\nabla f)(\bold{x}) = \sum_{i=1}^{n}(D_i f)(\bold{x}) e_i$

-이를 다시 풀어 적으면 아래와 같다.

$[D_1f_1(x), \\ D_2f_2(x), \\ \vdots \\ D_nf_n(x)]$

Directional derivative

-directional derivative of $f$ at $\bold{x}$, in the direction of the unit vector $\bold{u}$

$\lim_{t\to 0}\frac{f(\bold{x}+t\bold{u}) - f(\bold{x})}{t} = (\nabla f)(\bold{x})\cdot \bold{u}$

-can be denoted by $D_{\bold{u}}f(\bold{x})$

[Reference]

-PMA (Derivative in p.104, Partial derivative in p.215, Gradient in p.217, Directional derivative in p.218)