# strange step used in solutions

• Oct 17th 2009, 05:56 PM
walleye
strange step used in solutions
i wont write the whole question, but the term
$\displaystyle \frac{dy}{dx} \times \frac{1}{y}$
in the next line they change to
$\displaystyle \frac{d (ln(y))}{dx}$

i just dont see why you would be able to do that. none of the other terms in the equation (i havent written out) are changed. only thing i can think of is some sort of implicit differentiation idea or some sort. im not too sure
• Oct 17th 2009, 06:36 PM
mr fantastic
Quote:

Originally Posted by walleye
i wont write the whole question, but the term
$\displaystyle \frac{dy}{dx} \times \frac{1}{y}$
in the next line they change to
$\displaystyle \frac{d (ln(y))}{dx}$

i just dont see why you would be able to do that. none of the other terms in the equation (i havent written out) are changed. only thing i can think of is some sort of implicit differentiation idea or some sort. im not too sure

Without the whole question there is no context in which to judge why this is done. However, if you're question is that you don't understand the equality of the two expressions, then you should note that from the chain rule it follows that

$\displaystyle \frac{d (\ln y)}{dx} = \frac{d (\ln y)}{dy} \cdot \frac{dy}{dx}$ ....
• Oct 17th 2009, 06:38 PM
Mush
Quote:

Originally Posted by walleye
i wont write the whole question, but the term
$\displaystyle \frac{dy}{dx} \times \frac{1}{y}$
in the next line they change to
$\displaystyle \frac{d (ln(y))}{dx}$

i just dont see why you would be able to do that. none of the other terms in the equation (i havent written out) are changed. only thing i can think of is some sort of implicit differentiation idea or some sort. im not too sure

It is the chain rule of differentiation.

If I asked you to differentiate $\displaystyle \ln(y)$ with respect to $\displaystyle x$, you'd have to do it like so:

$\displaystyle \frac{d}{dx} \ln(y) = \frac{dy}{dx} \times \frac{d}{dy} \ln(y) = \frac{dy}{dx} \frac{1}{y}$
• Oct 17th 2009, 07:28 PM
walleye
ahhh i see, thanks to both of you

Quote:

Originally Posted by mr fantastic
Without the whole question there is no context in which to judge why this is done.

yeah its obvious (to me) why they do it, that step just confused me is all

thanks