Expert: Paul Klarreich - 10/14/2007

Question
seemingly simple differentiation giving me a head ache. the formula is   

arctan y/x = ln (x^2   y^2) ^1/2. it has to be differentiated as an implicit
function. the trying part is when i try differentiating all of the functions using
the
chain rule, it gets confusing. please help

Answer
Questioner:   Eliza
Category:  Calculus
Private:  No
 
Subject:  Calculus
Question:  seemingly simple differentiation giving me a head ache. the formula is  arctan y/x = ln (x^2   y^2) ^1/2. it has to be differentiated as an implicit function. the trying part is when i try differentiating all of the functions using the chain rule, it gets confusing. please help
............................................
Hi, Eliza,

Yes, you treat this as an implicit function.  But there are nice things about logarithms that you can use, such as:

log (AB) = log A + log B
log(A^n) = n log A

Here we go:

arctan(y/x) = ln (x^2   y^2) ^1/2

arctan(y/x) = 1/2 ln (x^2   y^2)

arctan(y/x) = 1/2 (ln x^2 + ln y^2)

arctan(y/x) = 1/2 (2 ln x + 2 ln y)

arctan(y/x) = ln x + ln y

Now differentiate.  The arctan(y/x) will take work, so do that separately.  It requires the chain rule AND the quotient rule.

=======================================================
WARNING: USE COURIER FONT TO VIEW THIS
===================================================

        (x)(dy/dx) - (y)(1)
D(y/x) = -------------------
             (x)^2

        x dy/dx - y
D(y/x) = -------------
            x^2
                      1
D(arctan(y/x))  =  ------------ D(y/x)
                  1 + y^2/x^2

                      1         x dy/dx - y
D(arctan(y/x))  =  ------------ --------------
                  1 + y^2/x^2       x^2

                   x dy/dx - y
D(arctan(y/x))  =  -------------
                   x^2 + y^2

Now put some stuff together.  The derivative of the right side is:

D(ln x + ln y) =  1/x + (1/y) dy/dx

So we can write the equation now:

x dy/dx - y      1    dy/dx
------------- = --- + ------
  x^2 + y^2      x      y

Move things around a bit:(get dy/dx terms on the left)

 x dy/dx     dy/dx        y          1
---------- - ------ = ----------- + ---
x^2 + y^2      y       x^2 + y^2     x

           x        1           y        1
dy/dx[ ---------- - --- ] = ----------- + ---
       x^2 + y^2    y       x^2 + y^2    x

       xy - (x^2 + y^2)      xy + (x^2 + y^2)
dy/dx[ ----------------- ] = ------------------
       (x^2 + y^2)y           (x^2 + y^2)x

       xy - x^2 - y^2       xy + x^2 + y^2
dy/dx[ ---------------- ] = ------------------
              y                    x

        (xy + x^2 + y^2)y
dy/dx =  ------------------
        (xy - x^2 - y^2)x

I think that's about it.