# 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 $$a(x,y,z)$$

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