I need help with a question in math involving differentiating a function

Q: Differentiate y = (cos x) / (1 - sin x)

Can someone explain in detail how to solve this?

You could use the quotient rule or the product rule.

If you define f(x) = cos(x) and g(x) = 1 / (1 - sin(x))

and use the product rule.

Product Rule

dy/dx [f(x)g(x)] = dy/dx[f(x)] * g(x) + dy/dx[g(x)] * f(x)

so

dy/dx[cos(x) * [1/(1 - sin(x))] ] = -sin(x) * [1/ [1 - sin(x)] ] +[ [-1 * (1 - sin(x))^-2 * -cos(x)] * cos(x) ]

= - sin(x) / [1 - sin(x)] + [ [cos(x)]^2 * (1 - sin(x))^-2] ]

= (-sin(x) / [1 -sin(x)] ) + ([cos(x)]^2 / [1 - 2*sin(x) + [sin(x)]^2])