Question

What is the implicit derivative of `y=(x-y)^2+4xy+5y^2`?

Answer 1

The implicit derivative is `dy/dx= (2x+2y)/(-12y-2x+1)`.

Explanation:

First, you would derive this by turning `y` into `dy/dx` when needed since it is a function while treating `x` as the variable. For the `(x-y)^2`, the chain rule is applied. This will result in:

`dy/dx=2(x-y)*(1-dy/dx)+4y+4xdy/dx+10ydy/dx`

For simplicity, we can distribute the values in the parentheses and then implement quadratic multiplication.

`dy/dx=(2x-2y)*(1-dy/dx)+4y+4xdy/dx+10ydy/dx`

`dy/dx=2x-2xdy/dx-2y+2ydy/dx+4y+4xdy/dx+10ydy/dx`

We now simplify like terms to get:

`dy/dx=2x+12ydy/dx+2y+2xdy/dx`

Move over all of the terms with `dy/dx` to the other side:

`-12ydy/dx-2xdy/dx+dy/dx=2x+2y`

Factor the `dy/dx` from the left side to get:

`dy/dx(-12y-2x+1)=2x+2y`

Divide the left and right sides by `(-12y-2x+1)` to achieve:

`dy/dx= (2x+2y)/(-12y-2x+1)`

There is your answer! Hope it helps!