2021-12-07

What is the difference between a derivative and a total derivative? Please, help.

Paineow

When you take a partial derivative, you operate under a sort of assumption that you hold one variable fixed while the other changes, while when computing a total derivative, you allow changes in one variable to affect the other.

Raymond Foley

So, for instance, if you have f(x,y)=2x+3y, then when you compute the partial derivative $\frac{\partial f}{\partial x}$, you temporarily assume y constant and treat it as such, yielding $\frac{\partial f}{\partial x}=2+\frac{\partial \left(3y\right)}{\partial x}=2+0=2$.
However, if $x=x\left(r,\theta \right)$ and $y=y\left(r,\theta \right)$, then the assumption that y stays constant when x changes is no longer valid. Since $x=x\left(r,\theta \right)$, then if x changes, it implies that at least one of r or$\theta$ changes. And if r or $\theta$ change, then y changes. Thus, obviously it has some sort of effect on the derivative and we can no longer assume it to be equal to zero.

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