Confusion with understanding derivatives in respect to

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Confusion with understanding derivatives "in respect to..."

I have been teaching myself Calculus for the past 2 weeks or so and I've just barely started learning Implicit Differentiation.

There's a few things I have trouble understanding, which this is probably a simple concept that I am failing to understand due to lack of teacher.

When taking the derivative of a something, with respect to something else, what does that mean?

All the videos and tutorials I find online mention, taking the derivative "with respect to x" or "with respect to y" or "with respect to b"... and this changes the answer(or does it?).

Can someone help me get an intuition on what it means to take the derivative with respect to something else and how it can change the answer? I'm not looking for any complex answers so if you wish to show an example, feel free to make it as easy as you want. I am just trying to understand what it means to take it with respect to something else. I know how to take derivatives on a fairly simple level, but I have not found any guides clarifying what "with respect to x or y" means.

Thanks.

 

[BAnders1]

A derivative can be thought of as a rate of change of a variable. The expression dx/dy in English is "the derivative of x with respect to y." What this means is that we are looking at the rate at which x changes when y changes.

In physics you might be interested in an object's speed. If the position of the object is a variable x, then the speed of the object is dx/dt (where t is time)-- that is, the speed of an object is the rate of change (derivative) in the object's position with respect to time."

 

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I have been teaching myself Calculus for the past 2 weeks or so and I've just barely started learning Implicit Differentiation.

There's a few things I have trouble understanding, which this is probably a simple concept that I am failing to understand due to lack of teacher.

When taking the derivative of a something, with respect to something else, what does that mean?

All the videos and tutorials I find online mention, taking the derivative "with respect to x" or "with respect to y" or "with respect to b"... and this changes the answer(or does it?).

Can someone help me get an intuition on what it means to take the derivative with respect to something else and how it can change the answer? I'm not looking for any complex answers so if you wish to show an example, feel free to make it as easy as you want. I am just trying to understand what it means to take it with respect to something else. I know how to take derivatives on a fairly simple level, but I have not found any guides clarifying what "with respect to x or y" means.

Thanks.

Suppose you have a function of more than one variable. Ie., f(a, b, c) = a²b + c.
You may not be interested in varying a general form of a vector in a, b, and c, ie., some of these may be constant and never vary. In that case you will want to specify that the derivative is with respect to either a, b, or c. Since we do not know if the other variables/constants are functions of the variable in question, we have to use the chain rule to get the most general form of the derivative.
Thus, if we were trying to find df/dc, we would see that df/dc = 2ab(da/dc) + a²(db/dc) + 1.

 

[HallsofIvy]

As the others have said, the "derivative of A with respect to B" measures how fast A is changing compared to how fast B is changing. Essentially it is "rate of change of A divided by rate of change of B".