## Monday, July 22, 2013

### Math Monday: Polynomial Functions and the Derivative—the Sum Rule

Posted by Rebekah

"If the ax is dull,
And one does not sharpen the edge,
Then he must use more strength;
But wisdom brings success." Ecclesiastes 10:10

As we have seen in previous posts, “sharpening” our algebra has indeed helped us calculate complex expressions using little “strength”.

But what about polynomial functions such as f(x) = 2x² + 8x + 5 ?

Well, let’s generalize the question and rephrase it as:

“But what about polynomial functions such as q(x) = f(x) + g(x) ?”

Using our definition of the derivative, we get…

Hey, look! We’re actually adding together two derivatives!

This then means that the derivative of q = the derivative of f + the derivative of g! Written in mathematical notation, this could be written as  $\frac{dq}{dx}=\frac{df}{dx}+\frac{dg}{dx}$

An alternative way of writing this would be q'(x) = f'(x) + g'(x)

Important note: The ' (i.e. prime) symbol is commonly used when referring to derivatives. In other words, f'(x) is the same as the derivative of f(x). Please remember this, as it will come in handy in the near future.

This “new” rule is called the sum rule.

Now we can solve the problem above!

The derivative of 2x² + 8x + 5 must equal 4x + 8   !

Important note: Why does the 5 “disappear”? Well, from previous studies in algebra, it should be clear that the derivative of a constant (e.g. 5) should be equal to 0.  This is because a derivative is a kind of slope, and if you have a function of any constant, the slope will always be 0 (this should be common algebra knowledge).   Furthermore, this can also be verified using the basic definition of the derivative, $\underset{\mathbf{d}\mathbf{x}\to \mathbf{0}}{lim}\frac{\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{+}\mathbf{d}\mathbf{x}\mathbf{\right)}\mathbf{-}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}}{\mathbf{d}\mathbf{x}}$; hence, lending further credibility to our power rule (click here to see the previous post on exponential functions).