and in this quite simple case, it is easily seen that the derivative
of a product is **NOT** the product of the
derivatives.
Although this naive guess wasn't right, we can still figure out what
the derivative of a product must be. Remember: When intuition fails,
apply the definition. Consider

A good way to remember the product rule for differentiation is ``the first times the derivative of the second plus the second times the derivative of the first.'' It may seem non-intuitive now, but just see, and in a few days you'll be repeating it to yourself, too.

Another way to remember the above derivation is to think of the
product `u`(`x`)`v`(`x`) as the
area of a rectangle with width `u`(`x`) and height
`v`(`x`). The change in area is
`d`(`uv`), and is indicated is the figure below.

- We can use the product rule to confirm the fact that the derivative
of a constant times a function is the constant times the derivative of
the function. For
`c`a constant, Whether or not this is substantially easier than multiplying out the polynomial and differentiating directly is a matter of opinion; decide for yourself.

- If
`f`and`g`are differentiable functions such that`f`(2)=3,`f`´(2)=-1,`g`(2)=-5 and`g`´(2)=2, then what is the value of (`fg`)´(2)?

- With
what is

`g`´(`x`)? - If
`f`,`g`and`h`are differentiable, use the product rule to show thatAs a corollary, show that This is a special case of the chain rule. - Find
`dy/dx`where

watko@mit.edu Last modified February 11, 2003.