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