Derivative Question

Question

Need help going through this questions please.

1. For the equation given below, evaluate dx/dy at the point (1,1).

(3x - y)^4 + 3y^3 = 19

dy/dx at (1,1) =

Answer 1

I'm guessing you mean dy/dx, not dx/dy, but you could find either. To find dy/dx, differentiate both sides of the equation with respect to x using implicit differentiation:

4(3x - y)³(3 - dy/dx) + (9y²)(dy/dx) = 0.

Now solve for dy/dx:

12(3x - y)³ - 4(3x - y)³(dy/dx) + 9y²(dy/dx) = 0 ==> dy/dx = -12(3x - y)³/[-4(3x - y)³ + 9y²]

Now plug in x = 1 and y = 1 to this to get dy/dx at (1, 1).