What is the reverse chain rule?

The reverse chain rule, also known as the chain rule in reverse, is a method used to simplify the differentiation of composite functions. It is the reverse process of the regular chain rule.

In the regular chain rule, if we have a composite function y = f(g(x)), we differentiate it by first differentiating the outer function f'(g(x)) and then multiplying it by the derivative of the inner function g'(x). 

However, in some cases, it may be easier to differentiate a composite function using the reverse chain rule. The reverse chain rule states that if we have a composite function y = f(g(x)), we can differentiate it by first calculating the derivative of the original function y with respect to g, and then multiplying it by the derivative of g with respect to x.

Here is an example to illustrate the reverse chain rule:

Let’s say we have the composite function y = (sin(x^2))^3.

Using the reverse chain rule, we start by calculating the derivative of y with respect to the inner function g(x) = sin(x^2):

dy/dg = 3g^2

Next, we calculate the derivative of g(x) = sin(x^2) with respect to x:

dg/dx = cos(x^2) * 2x

Finally, we multiply the two derivatives:

dy/dx = (dy/dg) * (dg/dx) = (3g^2) * (cos(x^2) * 2x)

So, the derivative of y = (sin(x^2))^3 using the reverse chain rule is (3(sin(x^2))^2) * (cos(x^2) * 2x).

Get ready for your mock exams with A Level Maths Christmas Revision