We need the chain rule to calculate the derivative of functions which has a function as its argument. The generic expression of such a function is:

Let's imagine we are asked to compute the derivative of this:

Here, we'd have as the outer function and as the inner function.

We can compute such derivatives by juggling a bit with the definition of the derivative. However, armed with the chain rule, you can take the short cut, straight to a neat little formula:

Using Leibniz's notation, the rule can also be written like this:

A side note: some literature use the notation to mean . These two have the same meaning.

To get a feeling for why the chain rule is correct, let's consider a math professor, she strolls slowly through the university corridor at night, as the windows shake from an actual train traveling at 10 times her speed. The sky is suddenly lit by a shooting star, moving at 2000 times the speed of the train.

How much faster is the star, compared to the math professor?

We know the relative speed of the star compared to the train. We also know how fast the train is compared to the professor. Using Leibniz's notation for derivatives, we can write:

This is the chain rule. It's just a way of breaking the derivative into smaller, more manageable parts.

In the example, the velocities are constant. We could have done the calculation without knowing about the chain rule. However, as the derivatives get more complicated, it's equally valid and all the more powerful.

Let's have a look at the example we mentioned at the beginning. We take the derivative of step by step, keeping in mind that we defined and . We get:

Frequently, you will encounter functions with an inner function inside the inner function, or where the inner function is a product. Sometimes, the composition is even more complicated.

When this happens, do not despair. To take the derivative of a composed function, we apply the rules in sequence.

However, some initial confusion is common, and taking the derivative is an art that requires practice. The exercises are a good place to start for mastering this subject.