# Basic, but subtle, calculus question.

1. Dec 14, 2006

### jrrship

Suppose we have a function,

y=Cx

where C is a constant.

we take the derivative of both sides:

dy/dx = d/dx (Cx) = C

so dy/dx = C

Then, we multiply both sides by dx:

dy = Cdx

Then supposed we want to get back to the original equation via integration--we integrate the right side with respect to x, but what do we integrate the left side with rexpect to?

It seems it would have to be y, but how can we integrate one side with respect to x and the other with respect to y?

Thanks!!

2. Dec 15, 2006

### matt grime

This is where treating dx and dy as if they were fractions is hiding what's going on.

You do not mutiply by dx then integrate both sides, one wtrx, one wrt y by guessing which is which.

If you want to do it that way, then you just put integral signs infront of dy and Cdx, and then it tells you what to integrate with respect to on each side - that is what the dy and dx are saying.

Better, though is to ster with dy/dx =C, and integrate both sides with respect to x and use the chain rule on the LHS.