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:

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