find y' if sin^-1(xy)=cosy

Use implicit differentiation:

sin^-1(xy)=cosy
sin(cos(y))=xy
differentiate both sides with respect to x
(use chain rule):
cos(cos(y))(-sin(y))y'=y+xy'
Collect terms and solve for y'
y'=-y/(cos(cos(y))sin(y)+x)

To find y', we need to differentiate both sides of the equation with respect to x.

Using the chain rule, let's differentiate sin^(-1)(xy) and cosy separately.

1. Differentiating sin^(-1)(xy):

Let u = xy
Then, y = u/x

Now, differentiate with respect to x:
du/dx = y + x * (dy/dx)

Next, differentiate sin^(-1)(u) with respect to u:
d(sin^(-1)(u))/du = 1/sqrt(1 - u^2)

Using the chain rule, we have:
d(sin^(-1)(xy))/dx = (1/sqrt(1 - (xy)^2)) * (y + x * (dy/dx))

2. Differentiating cosy:

Let's use the derivative of cosine:
d(cosy)/dy = -sin(y)

Now, let's differentiate both sides of the equation:

d(sin^(-1)(xy))/dx = d(cosy)/dx
(1/sqrt(1 - (xy)^2)) * (y + x * (dy/dx)) = -sin(y) * (dy/dx)

Now, let's solve for y':

Multiply both sides by sqrt(1 - (xy)^2):
y + x * (dy/dx) = -sqrt(1 - (xy)^2) * sin(y) * (dy/dx)

Move x * (dy/dx) to the left side:
y = -sqrt(1 - (xy)^2) * sin(y) * (dy/dx) - x * (dy/dx)

Factor out (dy/dx):
y = ((-sqrt(1 - (xy)^2) * sin(y)) - x) * (dy/dx)

Finally, isolate y':
y' = y / ((-sqrt(1 - (xy)^2) * sin(y)) - x)

And that's y' expressed in terms of the given equation.

To find y', we need to differentiate both sides of the equation with respect to x.

Let's start by differentiating sin^(-1)(xy) with respect to x.

The derivative of sin^(-1)(u) with respect to u is 1/sqrt(1-u^2), where u is a function of x. So, using the chain rule, the derivative of sin^(-1)(xy) with respect to x is:

d/dx(sin^(-1)(xy)) = (1/sqrt(1-(xy)^2)) * d/dx(xy).

To find d/dx(xy), we can use the product rule. The derivative of the product of two functions u(x) and v(x) is given by:

d/dx(uv) = u * d/dx(v) + v * d/dx(u).

In this case, u(x) = x and v(x) = y. Therefore, d/dx(xy) = x * d/dx(y) + y * d/dx(x).

Now substitute all the derivatives we found back into the equation:

(1/sqrt(1-(xy)^2)) * (x * d/dx(y) + y * d/dx(x)) = cos(y).

Simplifying the equation further, we get:

x * d/dx(y) + y * d/dx(x) = sqrt(1-(xy)^2) * cos(y).

Now, let's solve this equation for d/dx(y):

d/dx(y) = (sqrt(1-(xy)^2) * cos(y) - y * d/dx(x)) / x.

So, in order to find y', you need to find the derivative of x with respect to x, substitute it into the above equation, along with the values of xy and cos(y), and then simplify the expression.