Derivative

Derivative

말 그대로 미분 값. 함수의 특정 point에서 독립 변수($x$)의 미소 변화에 따라 종속 변수($f(x)$)의 변화량.
“For any $y=f(x)$, expressed as a multiplier $\alpha$ to a tiny increment$\Delta x$ to obtain the increments $\Delta y$ to the output”
$\Delta y=\alpha \Delta x$

Scalar function : output = scalar

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Scalar function

$f:{\mathbb{R}}^n\to \mathbb{R}$

$n=1$

scalar → scalar 이니, $\alpha =\frac{dy}{dx}$ : scalar

$n>1$

vector → scalar 이니, $\alpha =[{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n]$ : vector
정리해보면, $\Delta x$ : vector, $\Delta y$ :scalar, $\alpha$ :vector
$\Delta y={\alpha }_1\Delta x_1+{\alpha }_2\Delta x_2+\cdots +{\alpha }_D\Delta x_D$
$\Delta y=\frac{\partial y}{\partial x_1}\Delta x_1+\frac{\partial y}{\partial x_2}\Delta x_2+\cdots +\frac{\partial y}{\partial x_D}\Delta x_D$

$\Delta y={\nabla }_{\mathrm{x}}y\Delta x$
where,
${\nabla }_{\mathrm{x}}y=[\frac{\partial y}{\partial x_1},\cdots ,\frac{\partial y}{\partial x_D}]$
여기서, gradient 정의:
$\text{grdient}:={\nabla }_{\mathrm{x}}^{\top }y$