Tutoring calculus, this might be the first derivative rule you need to emphasize. The tutor opens up the discussion….

The product rule is not the first derivative rule people learn, but it might be the first non-intuitive one.  The other candidate might be the chain rule, which I’ll cover in another post.

I confess that when I first saw the product rule, I wanted to doubt it.  While I agreed it must be true, it seemed to lack the elegance I expected from calculus. That was at the start of my degree, of course; a year later, I’d gotten used to it.

By itself, the product rule is useful.  Moreover, several elegant techniques in later courses are based on it.

The product rule is used to find the derivative of a product of functions; ie, two functions multiplied together.  One statement of it is

(uv)’=uv’+u’v

Let’s look at an example:

Find the derivative: (x^2-3)sinx

Solution:

We first need to identify the functions u and v that make the product. In this case, it seems obvious enough to decide

u=x^2-3

v=sinx

Then, following the product rule,

((x^2-3)sinx)’=(x^2-3)(sinx)’+(x^2-3)’sinx

Now we take the separate derivatives as indicated:

((x^2-3)sinx)’=(x^2-3)cosx + (2x)sinx

We expand:

((x^2-3)sinx)’=x^2cosx -3cosx + 2xsinx

So, it turns out, by the product rule, that the derivative of (x^2-3)sinx is x^2cosx -3cosx +2xsinx.

Really, the product rule is kind of appealing, if you give it a chance:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.