# Chain rule

## Summary and examples

When we want to differentiate a function we can use the table of derivatives of standard functions, sometimes in combination with the product and quotient rule. Sometimes this is not possible, see the following example: This function can not be differentiated easily by merely using the table and product or quotient rules. In such cases we can use the chain rule.

There are both a formal and informal version of the chain rule. We explain the formal way by the following example.

##### Example 1

Differentiate: This function is not in the table of derivatives of standard functions and the product rule is not applicable.

We apply the transformation: that is to say, we switch to another variable. Then we get: Now we apply the following rule (the chain rule): and get: and: And thus: We want the result in only the variable and get: ##### Example 2

Differentiate: If we define: we get: and this function is in our table with standard functions. Applying the chain rule yields: and, back to : The informal method works as follows. We look again at the previous functions. If we want to differentiate: we could naively apply the table: which would have resulted in: but this is wrong. To get the correct answer we have to multiply this preliminary result by the derivative of the function that took the place of in the function . In this case we have to multiply by the derivative of which is . Then the result is: In the second example the ‘naive differentiator’ would have got as the derivative: but this is not the right answer. Again, to get the correct derivative, we have to multiply this result by the derivative of which is . The correct answer is: A few more examples.

##### Example 3 In this example we differentiate the standard -function and multiply the result by the derivative of .

##### Example 4 Again the -function is the standard function and the derivative is . We have to multiply this result by the derivative of .

##### Example 5

Differentiate: In this example we need to differentiate a square of the function . This means that is the standard function and that is in place of .

First we differentiate: and multiply the result by the derivative of . Note that the chain rule is an extra method to existing methods: the derivatives of standard functions and the product and quotient rule. We can combine these methods to determine a derivative, see the following example.

##### Example 6

Differentiate: In this case the numerator is: and the denominator is: and thus we have: (chain rule) (table)

and thus the result is: 0