Implicit Differentiation?

Question

Out of 80 problems these are the only couple I can't get so I figure I'm doing good Any help on these would be greatly apreciated since there all due tommrow.

Thanks


Find dy/dx by implicit differentiation


1.) cot(y)= x-y


2.) x=sec(1/4)

Answer 1

cot(y)= x-y

(-csc^2(y))*y' = 1 - y'
(-csc^2(y))*y' + y' = 1
y' (-csc^2(y) + 1) = 1
y' (-cot^2(y)) = 1
y' = -1 / cot^2(y)
y' = -tan^2(y)


x=sec(1/4)
This is the equation of a vertical line and therefore is not differentiable.

Answer 2

#1: just differentiate both sides, and solve for dy/dx:

-csc² (y) dy/dx = 1 - dy/dx
dy/dx - csc² (y) dy/dx = 1
dy/dx (1-csc² (y)) = 1
dy/dx = 1/(1-csc² (y))
dy/dx = -1/(csc² (y) - 1)
dy/dx = -1/(cot² y)
dy/dx = -tan² y

#2: Here, dy/dx does not exist, because y is not a function of x (if you graph this, it's a vertical line at sec (1/4)). That's fairly unusual for a homework set, check to see that there is no typo.

And on a further note -- 80 problems? Damn that sucks.

Edit: added last three lines of the solution to the first problem - I didn't see that simplification the first time around.