# How to tell if a quantity is defined as a derivative or a ratio

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• etotheipi
In summary, the conversation discusses the difference between equations defined in terms of ratios and derivatives. It is noted that while most equations without time involve ratios, there are exceptions such as dW/dx = F. The conversation also mentions a rule of thumb for determining if an equation is defined in terms of a derivative, but this rule is not always accurate. The conversation also delves into the definition of resistance and its relationship to V and I, with the conclusion that resistance can be defined in multiple ways and is not limited to just V/I. Finally, the conversation touches on the concept of defining equations in terms of different variables, such as density being defined as dM/dV for an object with varying density.
etotheipi
I appreciate that this is perhaps a strange question but it's been bugging me a little.

For instance, velocity is defined as the time derivative of position, so will always appear as the gradient a graph of x vs t. However, something like resistance as R = V/I is defined in terms of a ratio, and it would not be correct to interpret it as the gradient unless in the special case where current happens to be constant. Instead, we would determine R graphically by reading off values.

Is there an easy way to tell which is true just by looking at the formula or must they just be learned? One rule I've sort of made up is that when the equation includes time in some form, e.g. Q = It, it is likely actually defined in terms of a derivative like dQ/dt, since we're dealing with a rate of change. All other equations I can think of that don't include time seem to be defined with ratios, for instance things like V=IR, M=pV, F=PA etc.

This seems to work pretty well, but I don't want to learn something wrong that might cause confusion in the future. Can you think of any exceptions to this?

Edit

I just remembered dW/dx = F as a counter example, for instance if we plotted the cumulative work done as a function of displacement. Evidently my rule is a bit wrong.

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etotheipi said:
However, something like resistance as R = V/I is defined in terms of a ratio, and it would not be correct to interpret it as the gradient unless in the special case where current happens to be constant. Instead, we would determine R graphically by reading off values.
Not true. We can and do use R values as the slope of the tangent to the curve. Other times we can use the average R over some region of V and I. R=dV/dI is also useful at times. So R=V/I is only one of several definitions for R.

But in general, any derivative is just the ratio of infinitesimal differences. So I would not put such great importance on ratios versus derivative.

One amusing quote I remember from a teacher. "Differential calculus is just fun and games with subtraction. Integral calculus is just fun and games with addition."

etotheipi
anorlunda said:
Not true. We can and do use R values as the slope of the tangent to the curve. Other times we can use the average R over some region of V and I. R=dV/dI is also useful at times. So R=V/I is only one of several definitions for R.

But in general, any derivative is just the ratio of infinitesimal differences. So I would not put such great importance on ratios versus derivative.

One amusing quote I remember from a teacher. "Differential calculus is just fun and games with subtraction. Integral calculus is just fun and games with addition."

However, there does seem to be a material difference. If we take a displacement time graph which, for instance, might take the form of a sine wave. If we pick one point on the curve and measure the gradient, we obtain the velocity. If we pick another point with the same gradient, perhaps one cycle later, it will also give the same velocity. Though evidently the quantity x/t will be different for both points.

This causes me a bit of confusion when it comes to things like Ohm's law. The definition of resistance in my spec is the ratio of values read off the graph, though myself and many of my friends wouldn't necessarily know this just given the formula as gradients are so common for physics exams.

If we plot a graph of mass against volume for an object whose density is perhaps varies with volume, we would obtain the values of density by reading off the graph since density is defined as the constant of proportionality, so a gradient method (which might work in the case of constant density) would be wrong.

This is what's bothering me a little

etotheipi said:
The definition of resistance in my spec is the ratio of values read off the graph
You don't have to insist that the values go back to the origin.

For a simple single-valued curve, nearly linear it doesn't make much different. But there is no law that forbids a V-I curve that is many valued and almost any nonlinear shape. In a case like that, a line from the present V, I back to the origin is pretty useless. Differential R might be useful. But we could also solve a circuit containing such a device without ever defining any R, we just have V=f(I) or I=f(V).

Here's a silly example. No real-life device I know looks like that, but I'm just illustrating the generality of V versus I relationships.

Dale
anorlunda said:
You don't have to insist that the values go back to the origin.

For a simple single-valued curve, nearly linear it doesn't make much different. But there is no law that forbids a V-I curve that is many valued and almost any nonlinear shape. In a case like that, a line from the present V, I back to the origin is pretty useless. Differential R might be useful. But we could also solve a circuit containing such a device without ever defining any R, we just have V=f(I) or I=f(V).

Here's a silly example. No real-life device I know looks like that, but I'm just illustrating the generality of V versus I relationships.

View attachment 249963

I guess this makes sense, my definition of resistance is perhaps slightly arbitrary.

But on a completely different note, what if we consider that mass-volume equation. It would be silly to define density as dM/dV, for an object whose density might in this case vary with its volume. There are definitely some cases where an equation contains a constant of proportionality (though not necessarily "constant" per se) where it would be wrong to measure the gradient.

## 1. What is the difference between a derivative and a ratio?

A derivative is a mathematical concept that represents the instantaneous rate of change of a function at a specific point. It is calculated by taking the limit of the ratio of the change in the function's output to the change in its input as the change in input approaches zero. A ratio, on the other hand, is a comparison of two quantities expressed in terms of division.

## 2. How can I determine if a quantity is a derivative or a ratio?

If the quantity is expressed as the change in one quantity divided by the change in another quantity, it is most likely a ratio. However, if it is expressed as the limit of this ratio as the change in the second quantity approaches zero, it is a derivative.

## 3. Can a quantity be both a derivative and a ratio?

Yes, a quantity can be both a derivative and a ratio. This is because a derivative is a type of ratio, but not all ratios are derivatives. A quantity can be a ratio without being a derivative if it is not expressed as the limit of the ratio as the change in the second quantity approaches zero.

## 4. What are some real-life examples of derivatives and ratios?

Derivatives are commonly used in physics to represent velocity, acceleration, and other rates of change. In finance, derivatives are used to represent the rate of change of a financial asset's value. Ratios are used in everyday life, such as in cooking recipes, where ingredients are measured in ratios. They are also used in financial analysis to compare different companies' financial performance.

## 5. How can understanding derivatives and ratios be useful in daily life?

Understanding derivatives and ratios can help in making informed decisions in various fields such as finance, economics, and science. For example, knowing how to calculate and interpret derivatives can help in predicting future trends in the stock market. Understanding ratios can also be useful in everyday tasks, such as cooking and budgeting, where accurate measurements are essential.

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