Question

What is the implicit derivative of `1= xe^(4y`?

Answer 1

`-1/(4x)`

Explanation:

To find an Implicit Derivative, differentiate like normal except remember that when you differentiate a `y` there will be an extra `dy/dx` term, thanks to the Chain Rule.

`d/dx[1=xe^(4y)`]

Product Rule:

`0=e^(4y)d/dx[x]+xd/dx[e^(4y)]`

Find each derivative separately:

`d/dx[x]=1`

`d/dx[e^(4y)]=e^(4y)d/dx[4y]`

`d/dx[4y]=4dy/dx`

So, `d/dx[e^(4y)]=4e^(4y)dy/dx`

Plug them back in:

`0=e^(4y)+4xe^(4y)dy/dx`

`-e^(4y)=4xe^(4y)dy/dx`

`dy/dx=(-e^(4y))/(4xe^(4y))=-1/(4x)`