Use implicit differentiation to find dy/dx if cos xy = 2x^2 - 3y.
I'm stuck on this problem because I'm getting thrown off on how to factor this. Here's my work so far:
-{[(dy/dx)y + y]sin xy} = 4x - 3(dy/dx)
-{[(dy/dx)y + y]sinxy} + 3(dy/dx) = 4x
...now what?
Thx
To find the derivative dy/dx using implicit differentiation, you are on the right track. Here's how you can proceed from where you left off:
1. Start with the given equation: cos(xy) = 2x^2 - 3y.
2. Apply the chain rule to differentiate both sides of the equation with respect to x. Remember that y is a function of x, so you need to factor in the derivative of y with respect to x.
3. Differentiate the left side of the equation:
d/dx[cos(xy)] = d/dx[2x^2 - 3y]
-[(dy/dx)y + y]sin(xy) + cos(xy) * (dy/dx)(xy) = 4x - 3(dy/dx)
4. Simplify the equation by expanding the terms involving (dy/dx):
-[(dy/dx)y + y]sin(xy) + ycos(xy) * (dy/dx)x + xcos(xy) * (dy/dx)y = 4x - 3(dy/dx)
5. Group the terms involving (dy/dx) on one side of the equation:
-[(dy/dx)y + xcos(xy)](sin(xy) - 3) + ycos(xy)(dy/dx)x = 4x
6. Solve for dy/dx by isolating the term:
(dy/dx) = [4x + (dy/dx)y + xcos(xy)(sin(xy) - 3)] / [ycos(xy)]
At this point, dy/dx is still present in the equation, but we can isolate it on one side and simplify the equation further:
(dy/dx)[1 - x(cos(xy))(sin(xy) - 3)/y/cos(xy)] = 4x + x(sin(xy) - 3)/y
(dy/dx) = (4x + x(sin(xy) - 3)/y) / [1 - x(cos(xy))(sin(xy) - 3)/y/cos(xy)]
This is the final expression for dy/dx using implicit differentiation.