Required physics: none
Here are a few examples of using the chain rule for derivatives. The general form of the chain rule is:
Example 1. The chain rule applies when we need the derivative of a ‘function of a function’, as in where , so for the first example, we will make it explicit which is the first function and which is the second.
First (outer) function: . Second (inner) function: .
First take the derivative of the outer function with respect to its variable :
Now take the derivative of the inner function with respect to :
Finally, multiply the two together:
This example could have been done by substituting before doing the derivative and in that case the chain rule isn’t needed. That is we could have said
We could then have taken the derivative directly to get . The next example will illustrate a case where the chain rule must be used.
Example 2. First function: ; Second function: (so, effectively, ).
First, the outer derivative:
Then, the inner derivative:
Then, the combination
Notice in the last step, we substituted for in terms of .
Example 3. The quotient rule is often stated as a separate rule for finding the derivative of the quotient of two functions, such as . However, it is easily derived from the chain and product rules, as we’ll see here.
We can rewrite the quotient as a product:
We can now use the product rule to get
We now need to use the chain rule to find the last derivative in the above formula. If we define , then
where again we have replaced by in the last line, just to get everything back in terms of . We can now combine this result with the earlier one from the product rule to get:
which is the quotient rule.
Example 4. Find the derivative of . This is a quotient, but rather than use the quotient rule directly, we will treat it as a product and use the procedure from example 3 to find the derivative using the product and chain rules.
A good exercise for the reader is to check this result by using the quotient rule at the end of Example 3 directly and show you get the same answer.
Example 5. Find the derivative of . This is a four-level nested function, so we’ll need to use the chain rule several times. We can write this function as with , and . (You should take the time to write this out and verify that these substitutions work before proceeding.)
The chain rule can now be applied in the form
Seen broken down like this, the problem becomes much easier, since we need to work out four standard derivatives:
Combining all these, we get
Here, in the first line, we’ve just plugged in the derivatives as calculated above. Then in each succeeding line, we have worked backwards through the functions to put everything in terms of . The last line just rearranges a couple of things to make it neater.
The final result is complex, but there are no tricks involved; we just apply the chain rule in the normal way.