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Mathematics

Derivatives

Suppose we have a function as follows $f(x,h)$ , where $h(x)$ is also a function of $x$ .

Type of derivative	Notation	Meaning
Total derivative	$\frac{df(x,h)}{dx}$	How much $f$ changes when we vary the variable $x$
Partial derivative	$\frac{\partial f(x,h)}{\partial x}$	How much $f$ changes when we vary the argument of $f$ labeled $x$
Constrained derivative	$\left( \frac{ df(x,h) }{dx} \right)_{h}$	How much $f$ changes when we vary the variable $x$ with the constraint that $h$ is held constant.

The total derivative can be written in terms of partial derivatives using the chain rule.

$\frac{df(x,h) }{dx} = \frac{\partial f(x,h)}{\partial x} +\frac{\partial f(x,h)}{\partial h}\frac{\partial h(x)}{\partial x}$

Suppose we have a different function $g(x,y)$ , where $x(t)$ and $y(t)$ both are dependant on $t$ . Then

$\frac{d g(x,y) }{ dt } = \frac{\partial g}{\partial x} \frac{\partial x(t) }{\partial t} +\frac{\partial g}{\partial y} \frac{\partial y(t) }{\partial t}$

But $\frac{ \partial g(x,y) }{\partial t}$ is meaningless because $f$ does not have an argument labeled $t$ .

Sometimes the constrained derivative is the same as a partial derivative. For example,

$\left( \frac{df(x,h)}{dx} \right)_h = \frac{\partial f(x,h)}{\partial x}$

But another function $a(x,y,z)$

$\left( \frac{d a(x,y,z)}{dx} \right)_y = \frac{\partial a(x,y,z)}{\partial x} +\frac{\partial a(x,y,z)}{\partial z} \frac{\partial z}{\partial x}$