Question

Could someone explain this to me?

Answer 1

This is a simple differentiation problem. Let's start by taking the derivative of both sides (with respect to x):
$5y^4\frac{dy}{dx}+6y^2x+6x^2y\frac{dy}{dx}+20x^3 = 0$

Simplify:
$\frac{dy}{dx} (5y^4+6x^2y) + 6y^2x+20x^3 = 0$

Solve for dy/dx:
$\frac{dy}{dx} = \frac{-6y^2x-20x^3}{5y^4+6x^2y}$

Now, plug in the given points:
$\frac{dy}{dx} = \frac{-6(2)^2(-1)-20(-1)^3}{5(2)^4+6(-1)^2(2)} = \frac{24+20}{80+12}$

Further simplification gives:
$\frac{dy}{dx}|_{(2,-1)} =\frac{44}{92} = \frac{11}{23}$

So, your answer is 11/23 or B.