Some of my professors claim it’s the infinitismal section of the graph, and some say it’s just notation. Then my professor said the differential of a function and the derivative of a function are two different things, whatever that means. I’m just all very confused. Plz help.
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I’m going to elide over the history of mathematics and the calculus and all of that stuff and just cut to the quick:
Both sets of professors are right. There is a deeper meaning to d/dx where it does actually refer to the actual change in x, but *for your purposes,* you can treat it as notation. You’re integrating what’s between the integral symbol and the d/dx notation.
>Then my professor said the differential of a function and the derivative of a function are two different things, whatever that means.
Your professor is right! The derivative of a function measures *the rate at which the function is changing*, and the differential of the function *measures how much it actually changed.* For most purposes, in most cases, the difference is purely academic, but there are cases where it matters.
> [Why is it useful](https://math.stackexchange.com/questions/23902/what-is-the-practical-difference-between-a-differential-and-a-derivative/23914#23914) to have the distinction? Because sometimes you want to know how something is changing, and sometimes you want to know how much something changed. It’s all nice and good to know the rate of inflation (change in prices over time), but you might sometimes want to know how much more the loaf of bread is now (rather than the rate at which the price is changing). And because being able to manipulate derivatives as if they were quotients can be very useful when dealing with integrals, differential equations, etc, and differentials give us a way of making sure that these manipulations don’t lead us astray (as they sometimes did in the days of infinitesimals).
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