Meaning of partial differentiation

Question

Take a simple example:

$$\phi=xy$$

I know this is a stupid question but I was wondering if we are going to do partial differentiation w.r.t x why we cant just write $\frac{d\phi}{dx}$ instead of $\frac{\partial\phi}{\partial x}$, I think they should yield the same result?

Answer 1

No,

$$\dfrac{d\phi}{dx}=y+x\dfrac{dy}{dx}$$

whereas

$$ \dfrac{\partial\phi}{\partial x}=y$$

In the second case we are holding $y$ constant so $\dfrac{dy}{dx}=0$ but in the first case we are not holding $y$ constant, so the value of $\dfrac{dy}{dx}$ must be considered.

Answer 2

No, these are two different notations:

• $ \dfrac{d f}{d x}$ denotes the total dervivative
• $ \dfrac{\partial f}{\partial x}$ denotes the partial dervivative