Derivatives: Product, Quotient, and Chain Rules

Derivatives: Product, Quotient, and Chain Rules

Author: c o

To introduce the product rule, quotient rule, and chain rule for calculating derivatives
To see examples of each rule
To see a proof of the product rule's correctness

In this packet the learner is introduced to a few methods by which derivatives of more complicated functions can be determined.

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Before starting with this packet, you should be comfortable with the concept of a derivative of a function.

You will need to know how to find the derivative of powers of x and of the sine function, for your convenience, the shortcuts follow:



Often we are able to write a function as a kind of combination of two simpler functions, and when we can, we'd like to know how this affects the derivative.  An example of such a function might be f(x) = 2x2sin(x), which could break down into component functions like this: g(x) = 2x2, h(x) = sin(x) so that f(x) = g(x)h(x).  

But how do we calculate the derivative of such a function? It turns out that mathematicians have discovered rules that help to simplify this task.  This packet states and gives examples of these rules, and even proves why one of them works.

The Rules

The Product Rule


The Quotient Rule


The Chain Rule

And now for some examples.


In this video we see an example of each of the Product Rule, the Quotient Rule, and the Chain Rule.

Proving The Product Rule

In this video we apply our knowledge of the definition of the derivative to prove the product rule.