Can someone explain implicit differentiation in Calculus 1?

Okay when you do implicit differentiation you're differentiating with respect to one variable... typically it's in respect of x... so if you took the derivative of y with respect to x you'd end up with dy/dx... so lets do a simple implicit differentiation...

With respect to x:

y^2 = 3x^2 + 2y

2y(dy/dx) = 6x + 2(dy/dx) (when you take the derivative of something with respect to another variable you have to tack on a dy/dx or whatever the variable is...

Another example...

With respect to y:

y^2 = 3x^2 + 5c^3 + 4e

2y = 6x(dx/dy) + 15c^2(dc/dy) + 4(de/dy)

Notice how whatever the variable is your deriving in respect to, it will always be d(whatever the variable is)...

I hope this helped out a bit... I know implicit differentiation can be a bit confusing...

Its called  " implicit "   because  we  usually  do  not  have a   "nice"     EXplicit  form  of  y=
   a  good  example is a  circle:
  x^2 +y^2 =25
   To  get  an  explicit  form    of y =  we   would  have to  break it into  two   functions,  neither of  which is   pretty,  and       find  y'   with   various  ?  forms ...  ugh!
   In    Implicit  differentiation,   we  know   a  relation  that    implies     this  circle : x^2 +y^2 =25
  and     there  are  4 steps  :

  a)  Differentiate  BOTh  sides   wrt  x,  using  chain  rule,  product rule,  etc
b)  collect  all  the  dy/dx  terms  on  one  side, all other terms  on  other
c)   factor out  the   dy/dx,   and  
d) Solve  for   dy/dx (  usually  by  dividing    by the   factor     in  c)

   I  usually  tell  MY  students  to include the  dx/dx term    even though it  becomes  1,  in  order to  make the  process  clear.
    Example :   What is the  slope of  the  circle ( above )  at (3,-4) ?
    Solution  :
  D(LS) = d(RS )
  so : d/dx (  x^2 +y^2)  =  d(25)/dx
 2x(dx/dx)  +2y(dy/dy)  =0  ( step  a)
 2y(dy/dx)  =-2x(dx/dx  ( step  b
    = -2x
   and  dy/dx  = -2x/2y   ( step  c,  d )

     and  now we  substitute in   ( x,y) = ( 3,-4)
   so   dy/dx = -2x/2y = -2(3)/2(4) = -3/4

 Advantages of Implicit :   faster,   simpler,  no  complex  forms, can  be used if  there is  NO  explicit  form ,    and   a " neater "  answer

   Disadvantages of implicit :  Need to  know product  rule,  chain  rule,   algebra   collecting, etc,  and  to   get  an  exact  numerical  answer  you   need to know  BOTH an  x  and  a  y  ON  the  curve . :