In the last section we learned to differentiate quickly
rather than having to take the limits. We also learned about the product
rule and quotient rule which
is used when we are finding the derivative of a function that is being
multiplied or divided, respectively. However, to be able to fully use the
these three rules on a more variety of functions, we must be knowledgeable
of the chain rule. The **chain rule** summarized mathematically
says:

You might be wondering that this is wrong from what we learned in the last section. However, the x is in boldface which means that x is not just a single variable but a more complex expression. The derivatives of the polynomials we took were correct but there was equal to , and the derivative of x is 1 which when multiplied by a number doesn't change the number. The derivatives using the product rule and quotient rule were also correct. True f(x) and g(x) were expressions more complex than just , but in this case f(x) and g(x) had exponents equal to 1(the of x would be 1, which doesn't change the value of the derivative. Let's try to illustrate the chain rule on an example:

Let's try using the chain rule on another expression:

Here = , and n = 100. = . Putting it all together we have:

If you're not sure whether to use the chain rule or not be on the safe side and use it. The chain rule works in all cases, but only works on certain cases. That concludes discussion on the chain rule.