Final answer:
To find (dy)/(dx), we differentiate both sides of the equation using implicit differentiation. After differentiating each term, we solve for (dy)/(dx) by simplifying the equation.
Step-by-step explanation:
To find (dy)/(dx) using implicit differentiation, we differentiate both sides of the equation with respect to x. Let's start by differentiating x^(2)y using the product rule:
(d/dx)(x^(2)y) = 2xy + x^(2)(dy/dx)
Next, differentiate xe^(y) using the chain rule:
(d/dx)(xe^(y)) = e^(y) + xe^(y)(dy/dx)
Now, differentiate sinx which is a function of x: (d/dx)(sinx) = cosx
Substitute these results back into the equation: 2xy + x^(2)(dy/dx) + e^(y) + xe^(y)(dy/dx) = cosx
Now, solve for (dy/dx):
x^(2)(dy/dx) + xe^(y)(dy/dx) = cosx - 2xy - e^(y)
dy/dx = (cosx - 2xy - e^(y))/(x^(2) + xe^(y))