Find dy / dx by implicit differentiation.
2sin(9x)cos(3y)=3
dy/dx=_____
To find the derivative dy/dx using implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x, treating y as a function of x.
d/dx[2sin(9x)cos(3y)] = d/dx[3]
Step 2: Apply the chain rule to differentiate the composite functions.
[2cos(9x)cos(3y)(9) - 2sin(9x)sin(3y)(3(dy/dx))] = 0
Step 3: Simplify the equation and isolate dy/dx.
[18cos(9x)cos(3y) - 6sin(9x)sin(3y)(dy/dx)] = 0
[18cos(9x)cos(3y)] = [6sin(9x)sin(3y)(dy/dx)]
(dy/dx) = [18cos(9x)cos(3y)] / [6sin(9x)sin(3y)]
Therefore, dy/dx = [18cos(9x)cos(3y)] / [6sin(9x)sin(3y)]