What is the Chain Rule?

The Chain Rule is used to differentiate composite functions. A COMPOSITE FUNCTION is a ‘function inside a function’ but they can take many forms. The challenge with The Chain Rule is not just using the rule, but knowing when it can be used in the first place. The following are a selection of examples of composite functions – as you can see there are many different types.

Notice that in each case there are two functions present – an ‘outer’ function and an ‘inner’ function. For each example these are the outer and inner functions:

The Chain Rule

The Chain Rule itself can be written in several different ways. Some of these are more useful than others, especially when it comes to applying The Chain Rule in different situations. But here is the key to mastering The Chain Rule . . . it’s just the derivative of the outer function multiplied to the derivative of the inner function. So a good way to remember The Chain Rule is:

(the derivative of the outer) x (the derivative of the inner)

Translating this intuitive notion into Mathematical form gives us the following. Let the outer function be f(x) and let the inner function be u – where u is a function of x. Then each composite function is in the form f(u) and the derivative, using The Chain Rule, is given by:

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An Important Point . . . .

Regardless of how well you understand and learn The Chain Rule, you still have to differentiate the outer and inner functions successfully. To do so, you have to learn and practice the rules for differentiating other types of functions. Common function types are Polynomials (which require The Power Rule), Trigonometric Functions, and Exponential & Logarithmic functions. Download our FREE DIFFERENTIATION FORMULA LIST and see our Differentiation Q&A post HERE.

The Solutions

Have a go yourself at finding the derivative of the above example functions. Remember to just take the derivative of the outer and inner function and multiply them together. Then check the solutions below or watch our YouTube VIDEO on those questions.