# Product rule in Newton notation?

• I
• volican
In summary, the correct way to write the product rule in Newton notation is to use the rule as expected with Leibniz notation and then replace ##dx/dt## with x dot and ##dy/dt## with y dot. Using ##(xy)'## is preferable to using ##\stackrel{\cdotp}{(xy)}## as it is easier to read and less likely to be mistaken for a mistake. However, if using all dots, it is recommended to use Leibniz notation throughout for consistency.
volican
What is the correct way to write the product rule in Newton notation (with the dots above)? It is the LHS I am abit confused with. Eg. Say you have d/dt(xy) would you just put dots above the x and y?

Use the rule as expected with leibnitz notation and then at the end replace ##dx/dt## with x dot and ##dy/dt## with y dot:

##d(x.y)/dt = dx/dt.y + x.dy/dt = \dot{x}.y + x.\dot{y}##

https://en.wikipedia.org/wiki/Product_rule

## \stackrel{\cdotp}{(xy)} ## but in this case it would definitely be better to write ##(xy)'## instead.
##\stackrel{\cdotp}{x}\stackrel{\cdotp}{y}## would mean ##\frac{d}{dt}x \cdot \frac{d}{dt}y\,.##

Thanks. Why would (xy)' be better to use than the (xy) with a single dot above? In the equation I am working with I have just used all dots, would it be ok to include a single (xy)' ?

In this case stay with the dots or use the Leibniz notation everywhere. The dot over more than one variable is simply hard to read. E.g. it could be mistaken by a failure in paper, or just be overlooked.

## 1. What is the product rule in Newton notation?

The product rule in Newton notation is a mathematical formula used to find the derivative of a product of two functions. It states that the derivative of the product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.

## 2. How is the product rule used in physics?

In physics, the product rule in Newton notation is used to find the acceleration of an object that is experiencing multiple forces. By taking the derivative of each force function and applying the product rule, the net acceleration of the object can be determined.

## 3. Can the product rule be applied to more than two functions?

Yes, the product rule in Newton notation can be extended to any number of functions being multiplied together. The general formula for this is: the derivative of the product of n functions is equal to the first function multiplied by the derivative of the product of the remaining (n-1) functions, plus the second function multiplied by the derivative of the product of the remaining (n-1) functions, and so on.

## 4. How does the product rule differ from the chain rule?

The product rule and the chain rule are two separate rules used to find derivatives. The product rule is used when finding the derivative of a product of two or more functions, while the chain rule is used when finding the derivative of a composite function (a function within a function). Both rules involve taking the derivative of each individual function and combining them in a specific way.

## 5. Are there any limitations to using the product rule in Newton notation?

The product rule can only be applied to functions that are differentiable, meaning they have a well-defined derivative at every point. Additionally, the rule can only be used for products of functions, not for other mathematical operations such as division or addition. It is also important to note that the product rule only applies to functions of one variable.

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