Product, Quotient and Chain Rules

Product Rule

The product rule is utilised when differentiating a function that is the product of two separate functions.
If a function, y is defined as the product of two other functions, u and v (hence y = uv), the derivative of y in relation to x (dy/dx or y’) can be expressed as: dy/dx = u(dv/dx) + v(du/dx).
It’s worth memorizing the formula to become efficient at applying the product rule quickly.

When a function involves the quotient or division of two functions, the quotient rule is employed.
If y equals the division of two functions of x, u and v (i.e., y=u/v), then the derivative is given by the formula: dy/dx = (v(du/dx) - u(dv/dx))/v².
Like the product rule, practising the application of the quotient rule is vital as it can be slightly more complicated due to its structure.

When dealing with composite functions, the chain rule is used. A composite function is a function composed of two functions such as f(g(x)).
The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function.
If y = f(g(x)) then the derivative dy/dx is dy/dx = f'(g(x)).g'(x).
Chain rule is one of the most fundamental techniques in calculus, and it often comes into play when differentiating a wide range of functions.

These rules are applied extensively when dealing with functions that are product or quotient of simpler functions, or when dealing with composite functions.
The application of product rule, quotient rule and the chain rule are essential in solving problems like finding the slope of a tangent to a curve at a particular point, or finding the rate of change of quantities.
They are also key in solving more complex differentiation problems in physics, economics, engineering and other fields.
Mastering these rules broadens the scope of functions you can differentiate and helps tackle a variety of calculus problems.