How do you implicitly differentiate 22=(y)/(x-e^y)?

1 Answer
Apr 28, 2017

color(red) (dy/dx = 22/(1+22e^y))

Explanation:

Implicit differentiation is basically done in cases where y cannot be explicitly written as a function of x.

In this case,

22=(y)/(x-e^y)

=> 22* (x-e^y) = y

=x-e^y=y/22

Differentiating both sides with respect to x,

dx/dx - (de^y)/dx = 1/22*dy/dx

Using chain rule to evaluate (de^y)/dx

=> 1 - (de^y)/dy*dy/dx = 1/22dy/dx

=> 1 - e^ydy/dx=1/22dy/dx

=> (1/22+e^y)*dy/dx = 1

=> (1+22e^y)/22*dy/dx = 1

=> color(red) (dy/dx = 22/(1+22e^y))