Question #66265

2 Answers
May 20, 2015

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)

May 20, 2015

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]}