Expert: Ahmed Salami - 10/20/2009

Question
Using implicit differentiation, find dy/dx if tanxy=x

I don't know how to even start. Thank you in advance

Answer
Hi Stephanie,
Lets start by letting v = xy
dv/dx = x(1.dy/dx) + 1.y
     = x(dy/dx) + y
Note that when we differentiate any variable, say y, with respect to x we add dy/dx
Now,
tan(xy) = x
tan(v) = x
differentiating both sides with respect to x,
(secēv)dv/dx = 1
(secēv)[x(dy/dx) + y] = 1
[secē(xy)][x(dy/dx) + y] = 1
x(dy/dx) + y = 1/secē(xy)
x(dy/dx) + y = cosē(xy)
x(dy/dx) = cosē(xy) - y
dy/dx = [cosē(xy) - y]/x

Its good to know the method of implicit differentiation. But there is, however, a simpler method using the method of partial derivatives, just so you know.

Regards.