Differentiating by two different variables -- when is it okay?

[SamRoss]

I'm working through the problems in Mary Boas's Mathematical Methods text. Here's how she began solving one problem...

"We take differentials of the equation 1/i + 1/o = 1/f (f=constant) to get
-di/i² - do/o² = 0."

So on the left side the first term was differentiated with respect to i and the second term was differentiated with respect to o. Why is it okay to differentiate these terms by different variables? I'm only used to differentiating by one variable at a time.

 

[andrewkirk]

I'm working through the problems in Mary Boas's Mathematical Methods text. Here's how she began solving one problem...

"We take differentials of the equation 1/i + 1/o = 1/f (f=constant) to get
-di/i² - do/o² = 0."

So on the left side the first term was differentiated with respect to i and the second term was differentiated with respect to o. Why is it okay to differentiate these terms by different variables? I'm only used to differentiating by one variable at a time.

In general it is not OK.
In this case we can justify the step by first differentiating both sides wrt $o$ to get
$$-\frac{\frac{di}{do}}{i^2}-\frac1{o^2}=0$$
then multiply both sides by $do$

 

[SamRoss]

In general it is not OK.
In this case we can justify the step by first differentiating both sides wrt $o$ to get
$$-\frac{\frac{di}{do}}{i^2}-\frac1{o^2}=0$$
then multiply both sides by $do$

I get it. Thanks very much!