Question

Question #66265

Answer 1

`4x^3 + lny^4 -4y^2=10`

`d/dx(4x^3 + lny^4 -4y^2) = d/dx(10)`

`12x^2 + 1/y^4 * 4y^3 dy/dx -8y dy/dx =0`

`12x^2 + 4/y dy/dx -8y dy/dx =0` Now, clear the fraction (mult by `y`)

`12x^2y + 4 dy/dx -8y^2 dy/dx =0`

`dy/dx = (12x^2y)/(8y^2-4) = (3x^2y)/(2y^2-1)`

Answer 2

I assume lny^4=`ln(y^4)`, otherwise, you should have written `ln^4(y)`, but please, re-write the question LaTeX-ly

First of all, we need to write the gradient of `F(x)=4x^3 + 4ln(abs(y))-4y^2-10`, which is well defined `forall (x,y) : y!=0`

It is `(12x^2,4/y-8y)`, and it's always nonzero (trivial), so, for Dini's theorem, we can write `dy/dx` whenever `F_y!=0`, so whenever `y!=+-sqrt(2)/2` and it is defined as `dy/dx=-(F_x)/(F_y)=-((12x^2y)/(4-8y^2))`.

Notice that `F_x(0)=0` and `F(x,-)` is even `forall x`, so you have `y_1=y(x)` and `y_2=-y(x)` for the implicit function theorem.

graph{4x^3 - ln(y^4) -4y^2=10 [-1.709, 1.709, -0.853, 0.856]}