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

Question

Paineow · Accepted Answer

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 · Accepted Answer

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 (3 y)}{\partial x} = 2 + 0 = 2

.
However, if

x = x (r, θ)

and

y = y (r, θ)

, then the assumption that y stays constant when x changes is no longer valid. Since

x = x (r, θ)

, then if x changes, it implies that at least one of r or

θ

changes. And if r or

θ

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.

What is the difference between a derivative and a total

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