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The product rule is a rule of differentiation which states that for product of differentiable function's :

In prime notation:

In the case of three terms multiplied together, the rule becomes

It is one of the most common differentiation rules used for functions of combination, and is also very simple to apply. For instance, consider the function . The derivative is easily found:


Both of the following limits exist, as are differentiable

Then from first principles

Adding and subtracting a changes nothing, but allows for algebraic manipulation

Now, factoring, separating the fractions and distributing the limits with the algebra of limits yields

We can then use turn both of these into a product of limits and it is easy to see that

This completes the proof.