This is a simple differentiation problem. Let's start by taking the derivative of both sides (with respect to x):
[tex]5y^4\frac{dy}{dx}+6y^2x+6x^2y\frac{dy}{dx}+20x^3 = 0[/tex]
Simplify:
[tex] \frac{dy}{dx} (5y^4+6x^2y) + 6y^2x+20x^3 = 0[/tex]
Solve for dy/dx:
[tex] \frac{dy}{dx} = \frac{-6y^2x-20x^3}{5y^4+6x^2y} [/tex]
Now, plug in the given points:
[tex]\frac{dy}{dx} = \frac{-6(2)^2(-1)-20(-1)^3}{5(2)^4+6(-1)^2(2)} = \frac{24+20}{80+12}[/tex]
Further simplification gives:
[tex]\frac{dy}{dx}|_{(2,-1)} =\frac{44}{92} = \frac{11}{23} [/tex]
So, your answer is 11/23 or B.
