Find dy/dx by implicit differentiation.
cot(y) = 4x − 6y
-csc²y y' = 4 - 6y'
y' = 4/(6-csc²y)
To find dy/dx by implicit differentiation, follow these steps:
1. Differentiate both sides of the equation with respect to x. Treat y as a function of x.
We have cot(y) = 4x - 6y.
Differentiating cot(y) with respect to x gives -csc^2(y) * (dy/dx).
Differentiating 4x - 6y with respect to x gives 4 - 6(dy/dx).
2. Simplify the equation obtained.
The equation becomes -csc^2(y) * (dy/dx) = 4 - 6(dy/dx).
3. Solve for dy/dx.
Start by isolating the term with dy/dx by moving it to one side and the other terms to the other side.
We get -6(dy/dx) + csc^2(y) * (dy/dx) = 4.
Factor out (dy/dx) on the left side: (csc^2(y) - 6) * (dy/dx) = 4.
Now solve for dy/dx by dividing both sides by (csc^2(y) - 6):
dy/dx = 4 / (csc^2(y) - 6).
So, the derivative dy/dx is given by dy/dx = 4 / (csc^2(y) - 6).