What is the quotient rule and how is it applied?

The quotient rule is a formula used to differentiate fractions that contain terms of x in both the numerator and the denominator (we usually refer to this as f(x)/g(x)).
As it isn't possible to differentiate this by normal methods straight away, we use the quotient rule to allow us to do so. For A-level maths, you only need to know how to use the quotient rule, deriving how it is formed is not necessary. The quotient rule takes the following form:
dy/dx = ((g(x)f'(x) - f(x)g'(x))/g(x)^2
Where f'(x), g'(x) are dy/dx of f(x), g(x) respectively. Therefore differentiating the numerator and the denominator seperately, and then plugging these back into the formula, will yield the result of differentiating a fraction of the form f(x)/g(x).