1

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?

Alana
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2 Answers2

3

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.

John Wayland Bales
  • 21,814
  • thanks for answering, but if $y$ is not in term of $x$ then $\frac{d\phi}{dx} = \frac{\partial\phi}{\partial x}$ ? – Alana Nov 20 '16 at 17:38
  • The issue of partial differentiation arises only in the context of functions of several variables. If $y$ is a constant and not a variable, then one would not write $\phi(x,y)$, one would write $\phi(x)$ and $\dfrac{d\phi}{dx}$. It's also important when $x$ and $y$ are, in turn, functions of other variables. For example, let $\phi(x,y)=x^2y$, $x=2st, y=3t$. Then, using the chain rule, we have $\dfrac{\partial\phi}{\partial t}=\dfrac{\partial\phi}{\partial y}\cdot\dfrac{dy}{dt}$ but $\dfrac{\partial\phi}{\partial t}=\dfrac{\partial\phi}{\partial x}\cdot\dfrac{\partial x}{\partial t}$ – John Wayland Bales Nov 20 '16 at 18:15
  • Ran out of space, but notice that in the first of the last two equations we use $\dfrac{dy}{dt}$ because $y$ is a function of a single variable. But in the last equation we use $\dfrac{\partial x}{\partial t}$ because $x$ is a function of two variables. – John Wayland Bales Nov 20 '16 at 18:18
0

No, these are two different notations:

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