Chain Rule in Differentiation

Chain rule is a crucial rule in calculus used to differentiate a function that is a composition of one or more functions.
It is used when differentiating composite functions; functions within functions. For instance, a function expressed as f(g(x)).
In simple terms, the chain rule states that the derivative of a composition of functions is the derivative of the outer function multiplied by the derivative of the inner function.
The general formula is expressed as (f(g(x)))’ = f’(g(x)) * g’(x) where f(x) is the outer function and g(x) is the inner function.
To use the chain rule effectively, identify the ‘inner function’ and the ‘outer function’. Differentiate these separately and then multiply the results together.
For example, if you have a function y = (2x+3)^4, to differentiate, identify the inner function as (2x+3) (which has a derivative of 2) and the outer function as u^4 (which has a derivative of 4u^3). Applying the chain rule would give dy/dx = 4(2x+3)^3 * 2.
The chain rule can be used in conjunction with other differentiation rules such as the product rule and quotient rule, which means it’s not only limited to composite functions.
Understanding the chain rule is also crucial in finding the derivatives of trigonometric, logarithmic, and exponential functions.

Remember, practising the chain rule on various functions is the most effective way to internalise the concept. It can be challenging but critical for solving complex differentiation problems.