Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy for y*sec(x) = 5*x*tan(y).

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justaguide eNotes educator| Certified Educator

Given the expression `y*sec(x) = 5*x*tan(y)` the value of `dx/dy` has to be determined. Using implicit differentiation

`sec(x) + y*sec x*tan x*(dx/dy)` = `5*tan y*(dx/dy) + 5*x*sec^2 y`

=> `(dx/dy)(y*sec x*tan x -5*tan y) =5*x*sec^2 y - sec x`

=> `(dx/dy) = (5*x*sec^2 y - sec x)/(y*sec x*tan x -5*tan y)`

The derivative `dx/dy = (5*x*sec^2 y - sec x)/(y*sec x*tan x -5*tan y)`