How would I answer this question?

Question

Use the process of implicit differentiation to find dy/dx given that:

5x³y⁴ + 3y²/4x³ = 5xy

Apply directly without first simplifying the equation.

thank you.

* edit:

I tried answering the question but I got my final answer as (20x^3y^4 + 3/32x^-3y - 5x) · dy/dx = 9/64x^-4y^2 - 15x^2y^4 + 5y

is this correct? if not, what did I get wrong? how can I improve?

Answer 1

Hi Anne B.,

Let's see. The key here is that y is an implicit function of x, so you have to use the Chain Rule throughout to disentangle the variables, and break them down to "particles".

5x³y⁴ + 3y²/4x³ = 5xy : you will need Chain Rule for Products (two instances), and Chain Rule for Quotients

5(3x²y⁴ + 3x³y³y') + 3((4x³(6yy')-3y²(12x²))/(16x⁶) = 5(xy'+y)

Let me expand that, then condense:

15x²y⁴ +15x³y³y' +(9/2)x^-3y y' - (27/4)y²x^-4 = 5xy' + 5y

I don't know that that's what you would consider simplest form; if you need to collect all the non-y' terms and all the y' terms and express just y', I suppose you could. And/or multiply through by x⁴ to clear all the negative "x"ponents (so to speak).

-- Cheers, --Mr. d.

P.S. You would never be able to solve an equation like this analytically, the powers and non-separation of variables guarantee that. But, given an expression for y' = f(x,y) you could start graphing a y' vector field, indicating the paths of the multiple function line solutions. It will look bizarre, but so what.

At least, that's what I would do. It's been 50 years, maybe I'm rusty?

-- Cheers, --Mr. d.