Another implicit differentiation question

[donjt81]

ok so i did this problem but i wasnt sure if this is correct.

y = 8x + 5cos(xy) + 7

dy/dx = 8 + 5(-sin xy)(x dy/dx + y)
dy/dx = 8 - (5x) (sin xy) dy/dx - (5y) (sin xy)
(5x) (sin xy) dy/dx + dy/dx = 8 - (5y) (sin xy)
[(5x) (sin xy) + 1] dy/dx = 8 - (5y) (sin xy)
dy/dx = [8 - (5y) (sin xy)]/[(5x) (sin xy) + 1]

does that look right?

 

[drpizza]

Looks fine. I don't see any errors.

I wish I could show you via this forum how to "think" to go from the first line to the last line without any of the in-between steps.

Maybe you can figure it out from this: you know that you're going to get all the dy/dx terms on one side and then factor out dy/dx later. So, the numerator simply becomes all the non dy/dx terms, and the denominator becomes what's left. Minor attention to signs, and you save a LOT of writing.