Expert: Abe Mantell - 5/26/2009

Question
How do I do this problem where xy are together?

find dy/dx in terms of x and y if 2xy+8x+y=11??  

Answer
Well, in this case we could solve for y in terms of x, then easily find
dy/dx...but if the instruction is NOT to do that, then we use implicit
differentiation, as follows.

[2xy+8x+y]'=11'
where the prime (') indicates differentiation with respect to "x"
then we get
(2xy)'+(8x)'+y'=0
2(xy)'+8+y'=0, now use the product rule for (xy)'
2(xy'+y)+8+y'=0, now solve for y'
2xy'+2y+y'=-8
2xy'+y'=-8-2y
(2x+1)y'=-8-2y
Thus, y'=(-8-2y)/(2x+1)

Abe