sin y = x^2 = y in an open interval 0¡Ü x ¡Ü 2¦Ð find dy/dx

To find dy/dx, we need to take the derivative of the given equation with respect to x.

The given equation is sin(y) = x^2.

Step 1: Differentiate both sides of the equation with respect to x.

d/dx(sin(y)) = d/dx(x^2).

Step 2: Apply the chain rule on the left-hand side.

cos(y) * dy/dx = 2x.

Step 3: Solve for dy/dx by dividing both sides by cos(y).

dy/dx = 2x / cos(y).

Now, the equation doesn't explicitly provide the value of y in terms of x. To further evaluate dy/dx, we need to find a relationship between x and y.

Since we are working with an open interval 0 ≤ x ≤ 2π, we can find y as follows:

sin(y) = x^2.

Taking the inverse sine (arcsin) of both sides, we have:

y = arcsin(x^2).

Now, we can substitute this relationship in the expression for dy/dx:

dy/dx = 2x / cos(arcsin(x^2)).

Note that cos(arcsin(x^2)) can be determined using the Pythagorean identity: cos(arcsin(u)) = √(1 - u^2).

Therefore, the final expression for dy/dx is:

dy/dx = 2x / √(1 - (x^2)^2).