# How do you use implicit differentiation to find dy/dx given #xe^y-y=5#?

##### 1 Answer

# dy/dx = (e^y)/(1-xe^y)#

#### Explanation:

When we differentiate

However, we cannot differentiate a non implicit function of

When this is done in situ it is known as implicit differentiation.

We have:

# xe^y-y=5 #

Differentiate wrt

# (x)(e^ydy/dx)+(1)(e^y) - dy/dx=0#

# :. xe^ydy/dx + e^y - dy/dx=0#

# :. e^y = dy/dx-xe^ydy/dx#

# :. (1-xe^y)dy/dx = e^y#

# :. dy/dx = e^y/(1-xe^y) #

**Advanced Calculus**

There is another (often faster) approach using partial derivatives. Suppose we cannot find

# (partial F)/(partial x) (1) + (partial F)/(partial y) dy/dx = 0 => dy/dx = −((partial F)/(partial x)) / ((partial F)/(partial y)) #

So Let

#(partial F)/(partial x) = e^y#

#(partial F)/(partial y) = xe^y-1 #

And so:

# dy/dx = -(e^y)/(xe^y-1) = (e^y)/(1-xe^y)# , as before.