Mathematics

Derivatives

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

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 $a(x,y,z)$

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