What Basic Derivative Rule Is Responsible For This?

[DocZaius]

If:

$$a(x)=b(x)-c(x)$$

Then:

$$\frac{da}{dx}=\frac{db}{dx}-\frac{dc}{dx}$$

Please note that the sum/difference rule is not directly invoked since it talks about a derivative of a sum/difference being equal to the sum/difference of the derivatives. It could very well be the basis for this logic, although I don't see it offhand. And of course, this is not a homework question. I came across this logic today and was surprised I could use it.

 

[LCKurtz]

It certainly is the sum-difference rule. When you compute a'(x) on the left you are asking for (b(x) - c(x))' on the right, which, by the difference rule is b'(x) - a'(x).

 

[DocZaius]

Ah yes! Thank you. All I had to do was write "d(b-c)" instead of "da" and I would have seen it immediately.