How do you implicitly differentiate 6=ylny-e^x-e^y?

1 Answer
Jan 8, 2016

dy/dx=e^x/(lny+1-e^y)

Explanation:

Find the derivative of each part.

d/dx(6)=0

Use the product rule and chain rule here.

d/dx(ylny)=y'lny+y(1/y)y'=y'lny+y'

d/dx(e^x)=e^x

Chain rule:

d/dx(e^y)=y'e^y

Thus, the derivative of the entire implicit equation is

0=y'lny+y'-e^x-y'e^y

Solve for y'.

e^x=y'(lny+1-e^y)

y'=e^x/(lny+1-e^y)=dy/dx