How do you find #dy/dx# by implicit differentiation given #e^y=x^2+y#?

1 Answer
Dec 16, 2016

Note that since #y = y(x)#, taking the derivative of the entire function with respect to #y# requires you to account for that fact using the chain rule.

#(df)/(dx) = (df)/(dy)(dy)/(dx)#

where #(df)/(dy)# is the derivative of, say, #e^y# or #y# in your given function.

Thus:

#e^y (dy)/(dx) = 2x + 1(dy)/(dx)#

#=> (e^y - 1)(dy)/(dx) = 2x#

#=> color(blue)((dy)/(dx) = (2x)/(e^y - 1))#

If you know what #y# is in terms of #x#, you can plug it back in to give you a function containing just #x#.