Given that y = cos(3x)cosec(5x), use the product rule to find dy/dx.

Write out the product rule: if y=f(x)g(x) where f and g are functions, dy/dx = f'(x)g(x) + f(x)g'(x)
Substitute in the expressions from the question:Therefore if f(x)=cos(3x) and g(x) = cosec(5x), f'(x) = -3sin(3x) and g'(x) = -5cosec(5x)cot(5x)
Solve the question: It follows that if y=f(x)g(x), then dy/dx = -3sin(3x)cosec(5x) - 5cos(3x)cosec(5x)cot(5x) or equivalently dy/dx = -3sin(3x)/sin(5x) - 5cos(3x)cos(5x)/sin^2(5x)