Expert: Paul Klarreich - 8/5/2009

Question
find dy/dx by implicit differentation...
x+ln y=x^2y^6

Answer
Questioner: Clint
Country: United States
Category: Calculus
Private: No
Subject: calculus with applications
Question: find dy/dx by implicit differentation...
x+ln y=x^2y^6
..................................
Hi, Clint,

Sorry that the question was rejected the first time -- I just clicked on the wrong part of the screen.

Use chain rule and product rules:

x + ln y = x^2y^6

1 + 1/y dy/dx = x^2(6y^5 dy/dx) + (2x) y^6  << diff. term by term

1 + 1/y dy/dx = 6 x^2 y^5 dy/dx + 2x y^6

1/y dy/dx - 6 x^2 y^5 dy/dx = 2x y^6 - 1   << put all dy/dx terms on left

(1/y  - 6 x^2 y^5) dy/dx = 2x y^6 - 1   << factor

dy       2x y^6 - 1
-- = ------------------   << divide
dx    1/y  - 6 x^2 y^5


dy       2x y^7 - y
-- = ------------------   << clear fractions -- each term times y.
dx    1  - 6 x^2 y^6

That's it.