Expert: Steve Holleran - 10/16/2007

Question
I don't fully understand how to differentiate this problem (solve for dy/dx).

x+tan(xy)=0

Answer
Hi Kaylee,

Okay, you have to make use of the Chain Rule and Product Rule here.  

Also, you have to be sure, when you are differentiating with respect to x, if you come to a "y" item, you have to multiply by a "dy/dx":

            d/dx [ x + tan(xy) = 0 ]

         1 + [sec^2(xy) * (x * dy/dx + y * 1)] = 0

The stuff in the brackets is the derivative of tan(xy).
The sec^2 is the deriv of tangent, and the stuff in the parentheses is the product rule on "xy".

Then you have:

 x * sec^2 (xy) * dy/dx + y * sec^2 (xy) = -1

and   

    x * sec^2 (xy) * dy/dx = -1 - y * sec^2 (xy)

dividing,

     dy/dx = [-1 - y * sec^2 (xy)] / [x * sec^2 (xy)]

I hope you could follow this okay.

Good luck,

Steve