# When Can You Manipulate Differentials Like Fractions?

*In general, you can manipulate total derivatives like fractions, but you can't do the same with partial derivatives.*

*Cross-posted from here.*

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If you have a function $f(x,y)$ where $x=x(t)$ and $y=y(t)$ are themselves functions of a parameter $t,$ and you blindly cancel out differentials, then you can get to incorrect statements like

whereas what’s actually true is

You can’t cancel because the $\partial f$’s in the numerators of $\dfrac{\partial f}{\partial t},$ $\dfrac{\partial f}{\partial x},$ $\dfrac{\partial f}{\partial y}$ all mean different things.

- The $\partial f$ in the numerator of $\dfrac{\partial f}{\partial t},$ represents the change in $f$
*attributed to the change in $t.$* - The $\partial f$ in the numerator of $\dfrac{\partial f}{\partial x}$ represents the change in $f$
*attributed to the change in $x.$* - The $\partial f$ in the numerator of $\dfrac{\partial f}{\partial y}$ represents the change in $f$
*attributed to the change in $y.$*

But in single-variable calculus, you’re working exclusively with functions that have only one input variable. And if you have a function $f(x)$ where $x=x(t)$ is itself a function of a parameter $t,$ then it’s true that

The above is conventionally written with “total” derivative symbols ($\mathrm d$ means “total”, $\partial$ means “partial”) since the change attributed to the single variable is the same as the total change of the function.

So in general, you can manipulate total derivatives ($\mathrm d$) like fractions, but you can’t do the same with partial derivatives ($\partial$).

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