Question

Alright so implicit differentiation is just not working out for me.

Use implicit differentiation to find the slope of the tangent line to the curve at point (4,1).

y / (x-2y) = x^3 + 4

Tried quotient rule to get the derivative of the left side, then got derivative of right side, and ultimately I got y + 3x^2 (x-2y)^2 / x-2y-2y

This is clearly not correct since the bottom would turn into zero if I plugged in x and y. Sigh...

Answer 1

It will be easier if you multiply it out and combine terms.
9y = x^4 +4x -2x^3 y
Differentiate both sides with respect to x implicitly, treating y as a function of x.
9 dy/dx = 4x^3 + 4 -2x^3 dy/dx
-6x^2 y
dy/dx(9 + 2x^3) = 4x^3 + 4 - 6x^2 y
dy/dx = [4x^3 -6x^2 y +4]/(9 + 2x^3)
Plug in x=4 and y=1. You should not get a zero deniminator.