Expert: Paul Klarreich - 1/1/2008

Question
My text book derives the differential of Arcsin(u/a) where a is a constant by:

d(Arcsin u/a) = [1/sqrt(1-(u/a)^2] [du/a] = [a/sqrt(a^2-u^2)] [du/a]

My question is how do we go from [1/sqrt(1-(u/a)^2]

To:  a/sqrt(a^2-u^2)

In other words, how did we arrive at getting an "a" in the numerator and getting "a" in sqrt(a^2-u^2) in the denominator.

Is this just a algebriac operation?

Thanks

Answer
Questioner:   Mark
Category:  Calculus
Private:  No
 
Subject:  Inverse Trig Forms (Differentials)
Question:  My text book derives the differential of Arcsin(u/a) where a is a

constant by:

d(Arcsin u/a) = [1/sqrt(1-(u/a)^2] [du/a] = [a/sqrt(a^2-u^2)] [du/a]

My question is how do we go from [1/sqrt(1-(u/a)^2]

To:  a/sqrt(a^2-u^2)

In other words, how did we arrive at getting an "a" in the numerator and getting

"a" in sqrt(a^2-u^2) in the denominator.

Is this just a algebriac operation?

Thanks
.............................................
Hi, Mark,

Yes.

OK, I suppose you want more than that --

1. Chain rule:

let v = u/a,  dv/du = 1/a
y = arcsin(v), dy/dv = 1/sqrt(1 - v^2)
                     1            1    
d(Arcsin(u/a) = ----------------- ---
               sqrt(1 - (u/a)^2)  a

2. Algebra:
                     1                
d(Arcsin(u/a) = --------------------
               a sqrt(1 - (u^2/a^2)

                     1                
d(Arcsin(u/a) = --------------------------
               sqrt(a^2) sqrt(1 - u^2/a^2)

                     1                
d(Arcsin(u/a) = ----------------
               sqrt(a^2 - u^2)

............FOLLOWUP.................
Hi, Mark,

I think I didn't REALLY answer your question last time.  I just appeared to.


1. Chain rule:

let v = u/a,  dv/du = 1/a
y = arcsin(v), dy/dv = 1/sqrt(1 - v^2)
                     1            1    
d(Arcsin(u/a) = ----------------- ---
               sqrt(1 - (u/a)^2)  a

2. Algebra:
First, the LCD inside is a^2
                     1            1    
d(Arcsin(u/a) = ----------------- ---
               sqrt(1 - (u^2/a^2)  a

                     1                 1    
d(Arcsin(u/a) = ---------------------- ---
               sqrt((a^2 - u^2)/a^2)   a


                     1                    1    
d(Arcsin(u/a) = ----------------------    ---
               sqrt(a^2 - u^2)/sqrt(a^2)  a

                     1                 1    
d(Arcsin(u/a) = ---------------------- ---
               sqrt(a^2 - u^2)/a       a

To divide fractions, invert denominator:

                     a             1    
d(Arcsin(u/a) = ------------------ ---
               sqrt(a^2 - u^2)     a

                      1
d(Arcsin(u/a) = ----------------
               sqrt(a^2 - u^2)