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 \(a(x,y,z)\)

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