A few weeks ago I was introduced to the chain rule in my Coursera/Mooculus course. And I found it to be one of those concepts that is straight forward to understand but hard to our into practice.
The chain rule deals with composition of functions. In other words, it deals with the derivative of functions within functions.
The best example I can think of to explain this is,the following: Imagine you are a salesperson whose job is to call customers, set up an appointment for a consult, give them a sales presentation and close a sale.
That sale depends on how many presentations are given, which are dependent of how many appointments were made, which are in turn dependent on how many calls were made. If you had a formula that described this process and you wanted to know how do a change in calls affect sales you might need to use the chain rule.
Let's imagine that such a formula exists. We will use the following variables for it: Sale (S), Presentation (P), Appointments (A), and Calls will be our (x). The formula is S(x)= P(A(x)), in order to know how changes in x affect S(x) we will need to differentiate the function with using the chain rule:
S'(x) = P'(A(x))(A'(x)) which is the same as saying we are taking the derivative of the outside function evaluated at the inside option and we multiply that with the derivative if the inside option.
Let's imagine than in the example above these formulas can be substituted for S(x)= √(3/4x). How will a change in x affect S(x)?
S'(x)= [1/2(3/4x)^-1/2](3/4) and simplifying we get:
Let's imagine the sales person makes 100 calls a day according to the original formula he will get approximately 8.66 sales. What if the sales person makes 50 extra calls by what amount would sales change?
According to the derivative sales would change .03535 times multiplied by the 50 extra calls which would yield approximately 1.76 extra sales. Now that's the way I understood it. But, since I might be wrong, here are five videos explaining the chain rule.
Professor Jim Fowler produced this amazing video explaining the concept. It is 10 minutes long but for someone strugling to understand what the chain rule is and how it works, its worth the time.
Here is the chain rule introduction by Khan Academy.
I liked this example from That Tutor Guy
Another straightforward example from justmathtutoring.com
This one is from the IntegralCalc channel in YouTube