Basic, but subtle, calculus question.

[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?

Please advise.

Thanks!

 

[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.

 

[tacman]

When we take the derivative of a function, we are essentially finding the rate of change of that function with respect to the variable x. In this case, we are taking the derivative of y with respect to x, so we are finding the rate of change of y with respect to x. This is represented by dy/dx.

On the other hand, when we integrate a function, we are essentially finding the area under the curve of that function. In this case, we are integrating the right side of the equation, which is Cdx, with respect to x. This means we are finding the area under the curve of Cx with respect to x.

So, to integrate the left side of the equation, which is dy, we need to integrate it with respect to y. This may seem confusing at first, but think of it this way: we are finding the area under the curve of y with respect to y. In other words, we are just finding the area of a rectangle with height y and width dy.

Therefore, to get back to the original equation, we would integrate both sides with respect to their respective variables. The integral of dy with respect to y is simply y, and the integral of Cdx with respect to x is Cx.

I hope this helps clarify the process of integrating both sides of an equation with respect to different variables. Remember, when taking derivatives and integrals, we are always considering the rate of change and the area under the curve with respect to a specific variable.