Question

What is the implicit derivative of `13=2xe^y-x^3e^(2y)`?

Answer 1

`dy/dx=(3x^2e^y-2)/(2x(1-x^2e^y))`

Explanation:

This will require Implicit Differentiation, Chain Rule, and Product Rule. The most important thing to remember is that since we're differentiating with respect to `x`, anything with `y` will spit out a `dy/dx` term.

`d/dx[13=2xe^y-x^3e^(2y)]`

Let's first find the derivative of each inner part.

`d/dx[2xe^y]=2d/dx[xe^y]=2(e^y+xe^ydy/dx)=2e^y+2xe^ydy/dx`

`d/dx[x^3e^(2y)]=3x^2e^(2y)+2x^3e^(2y)dy/dx`

Back to the original:

`0=2e^y+2xe^ydy/dx-3x^2e^(2y)-2x^3e^(2y)dy/dx`

`3x^2e^(2y)-2e^y=dy/dx(2xe^y-2x^3e^(2y))`

`dy/dx=(3x^2e^(2y)-2e^y)/(2xe^y-2x^3e^(2y))`

`dy/dx=(e^y(3x^2e^y-2))/(e^y(2x-2x^3e^y))`

`dy/dx=(3x^2e^y-2)/(2x(1-x^2e^y))`