Expert: Paul Klarreich - 3/9/2007

Question
I need help on a homework problem.  The directions say find dy/dx.   

xe^(y)+ 1= xy

I don't really know what i'm doing wrong.  I tried the problem a few different ways... i moved the x over so y would be by it self and took the derivative, then i left it and tried to take the derivative, but i'm having no luck.

Answer
Questioner:   Stephanie
Category:  Calculus
 
Subject:  calculus
Question:  I need help on a homework problem.  The directions say find dy/dx.   

xe^(y)+ 1= xy

I don't really know what I'm doing wrong.  I tried the problem a few different ways... I moved the x over so y would be by itself and took the derivative, then I left it and tried to take the derivative, but I'm having no luck.
...................................
Hi, Stephanie,

Many years ago, before you were born, no doubt, there was a black comedy movie with the title:  

Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb (1964)

So the title of this answer is : How you learned to love implicit differentiation and stop working so hard.

That's what it is -- an Implicit Differentiation exercise.  The process for I.D. is:

Step 1: Don't solve for y.  [Part of your grade depends on how well you don't do this.]

Step 2: Differentiate each term on both sides.  Whenever you have to differentiate something involving y, use the Chain Rule if needed; you will get a factor of dy/dx.  If dy/dx does not show up at least once,  (probably more than once) you did something wrong.

Step 3: Solve for dy/dx.  Your answer (probably) will have both x and y in it.  Don't worry about that, and don't worry about all the minus signs, either.

xe^(y) + 1  = xy

Differentiate the term:  x e^y.  Use the product rule and chain rule.

D(x e^y) = (x)(e^y dy/dx) + (1)(e^y) = x e^y dy/dx + e^y

D(1) = 0  << the second term.

Differentiate the term: xy.  Use the product rule.

D(xy) = (x)(dy/dx) + (1)(y) = x dy/dx + y

Put it all together:

x e^y dy/dx + e^y = x dy/dx + y

Solve for dy/dx:

x e^y dy/dx - x dy/dx = y - e^y

(x e^y  - x)dy/dx = y - e^y

dy     y - e^y
-- = ------------
dx    x e^y  - x

That is the answer, or you could factor the bottom.

dy     y - e^y
-- = -----------
dx   x(e^y  - 1)