A curve has equation:
3siny + 4cosy = xy + x^2
Q:find dy/dx in terms of x and y.
To find the derivative dy/dx in terms of x and y, we can use implicit differentiation. Implicit differentiation allows us to find the derivative of equations that are not explicitly given in terms of x and y.
Step 1: Differentiate both sides of the equation with respect to x. Treat y as a function of x and use the chain rule to differentiate the trigonometric functions.
Differentiating 3siny + 4cosy with respect to x gives us:
3cosy(dy/dx) - 4siny(dy/dx) = y + 2x
Step 2: Move terms involving dy/dx to one side of the equation and factor out the common factor:
(dy/dx)(3cosy - 4siny) = y + 2x
Step 3: Divide both sides of the equation by (3cosy - 4siny) to solve for dy/dx:
dy/dx = (y + 2x) / (3cosy - 4siny)
So, dy/dx in terms of x and y is (y + 2x) / (3cosy - 4siny).