Originally Posted by

**deltasalt** Hi,

Through a maths course I have been learning about differentiating inverse trig functions. I just learnt the rule:

$\displaystyle $y = {\sin ^{ - 1}}\left( {{x \over a}} \right)$$

$\displaystyle \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{a^2} - {x^2}} }}$

I understand how to use the rule and how to prove the rule.

I was recently shown a question $\displaystyle y = {\sin ^{ - 1}}\left( {\frac{x}{{{a^2}}}} \right)$ and to solve it they did

$\displaystyle \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{{\left( {{a^2}} \right)}^2} - {x^2}} }}$

$\displaystyle = \frac{1}{{\sqrt {{a^4} - {x^2}} }}$

And then continued making it more elegant

But what I don’t understand is why were they able to use the same rule and just sub it into the end?

I mean if you got

$\displaystyle y = {\sin ^{ - 1}}\left( {\frac{{{x^2}}}{a}} \right)$

You couldn’t just say that

$\displaystyle \frac{{dy}}{{dx}} = \frac{1}{{\sqrt {{a^2} - {{\left( {{x^2}} \right)}^2}} }}$

Because that would be wrong because you need to use chain rule.

So why can “a” be pretty much anything but “x” must remain as “x” for you to be able to use the rule?

I tried to ask my teacher and he said it is because “a” is a constant. That confused me even more because I am not sure what the difference is between a variable and an unknown constant. Seeing as they both can take on any value.

Do constants have some sort of property where you can replace them for anything in a rule and the rule still works while variables can’t be replaced without creating a new rule?

Sorry if this is worded badly I am having a bit of trouble trying to explain why I am confused. I would really appreciate any help.

Thank you