Properties of Logarithms: why d(ln y)/dx = (1/y)(dy/dx) ?

[Lime]

What property states that you can go from this:

ln y = 4 ln(2x - 3) + 5 ln(3x + 5) - 6 ln(5x + 4)

To this:

1/y dy/dx = 4(2 / 2x - 3) + 5(3 / 3x + 5) - 6(5 / 5x + 4)

Firstly, I see how on the left side, that 1/y is the derivative of ln y. But why must this be multiplied by dy/dx? According to the chain rule, dy/dx = dy/du * du/dx. How would dy/dx ever be multiplied by something?

Secondly, please explain the right hand side as well.

 

[Lime]

Actually I just figured out the right hand side.

But if someone could fill me in on the left hand side that would be appreciated.

 

[Steem]

When you take the derivative of y with respect to x you get dy/dx. On the left side of your problem you are taking the derivative of the ln y with respect to x which equals 1/y but because the derivative is taken with respect to x,you have to include dy/dx. It's the same as implicit differentiation.
Even when you take the derivative of x with respect to x you get dx/dx which is equal to 1. That's why you don't write it out.
Hope this helps!

 

[stapel]

I see how on the left side, that 1/y is the derivative of ln y. But why must this be multiplied by dy/dx?

Has your class covered the Chain Rule at all? (Since y is a function of x, and since you're differentiating y(x) with respect to x, naturally you'd have to end up with a "dy/dx" in there somewhere. But if you haven't yet seen the Chain Rule, then I'm not sure how you'd get that, is why I ask.)

Thank you.

Eliz.