Sum/Difference Rule

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We develop the rules needed to split functions across addition and subtraction signs, which allows us to take derivatives term by term.

Video Lecture

Text and Additional Details

Just like with limits, we want to develop some techniques that will allow us to break apart larger functions and expressions into smaller and easier to tackle pieces when taking derivatives. We will see that in some (but not all!) cases this will work the same for derivatives as it did for limits.

Remember that the derivative is just the limit of a difference quotient. So if we want the derivative of $f(x)$ at some $x$ value, say $b$ , then we want to calculate: $\lim \limits _{x\rightarrow b} \frac {f(x)-f(b)}{x-b}$ A common notation for wanting to compute a derivative of $f(x)$ is the following: $\frac {d}{dx} f(x) = f'(x)$ where $f'(x)$ is the derivative of $f(x)$ as normal.

But, what if we want to calculate the derivative of $f(x) + g(x)$ or $f(x) - g(x)$ ? What about $3\cdot f(x)$ ? Or $-7\cdot f(x)$ ? It helps to build the difference quotient for these expressions and then do some algebraic manipulation to see if, and how, we might use limit laws to make this easier!

First, let’s try and determine the derivative of $f(x) \pm g(x)$ at $x=b$ .

Note: The $\pm$ sign means plus or minus. If you aren’t sure how to deal with this symbol, or its reverse version $\mp$ , you can go through any computation by only looking at the top symbol, or by only looking at the bottom symbol. As long as you are consistent and always look at either the top or the bottom for the entire calculation, it will work. The point of the $\pm$ and $\mp$ symbols, is to show that the calculation works with either sign, $+$ or $-$ , simultaneously. This is standard mathematical notation and technique, worth taking a moment to familiarize yourself with for future work.

$\displaystyle \lim \limits _{x\rightarrow b} \frac {\left (f(x) \pm g(x)\right ) - \left (f(b) \pm g(b)\right )}{x-b}$	$\displaystyle = \lim \limits _{x\rightarrow b} \frac {f(x) \pm g(x) - f(b) \mp g(b)}{x-b}$	Distribute the negative
	$\displaystyle = \lim \limits _{x\rightarrow b} \frac {f(x) - f(b) \pm g(x) \mp g(b)}{x-b}$	Reorder terms
	$\displaystyle = \lim \limits _{x\rightarrow b} \frac {\left (f(x) - f(b)\right ) \pm \left (g(x) - g(b)\right )}{x-b}$	Factor out “ $-1$ ” from second pair of terms
	$\displaystyle = \lim \limits _{x\rightarrow b} \left (\frac {f(x) - f(b)}{x-b} \pm \frac {g(x) - g(b)}{x-b}\right )$	Split Difference Quotient
	$\displaystyle = \lim \limits _{x\rightarrow b} \left (\frac {f(x) - f(b)}{x-b}\right ) \pm \lim \limits _{x\rightarrow b} \left (\frac {g(x) - g(b)}{x-b}\right )$	Split Limit using Limit Law

On the last line we used the fact that the sum (or difference) of limits is the limit of the sum (or difference); i.e., we can split limits over sums and differences. This tells us that derivatives can be split over sums and differences, just like limits! In other words, the derivative of $f(x) + g(x)$ is the same as the derivative of $f(x)$ plus the derivative of $g(x)$ , and similarly for subtraction.

Now, how about a constant multiplier? Let $C$ be any real number, then can we use limit laws to simplify the process of taking the derivative of $C \cdot f(x)$ ? Let’s see!

$\displaystyle \lim \limits _{x\rightarrow b} \frac {Cf(x) - Cf(b)}{x-b}$	$\displaystyle = \lim \limits _{x\rightarrow b} \frac {C\left (f(x) - f(b)\right )}{x-b}$	Factor out $C$ on top
	$\displaystyle = \lim \limits _{x\rightarrow b} C \left (\frac {f(x) - f(b)}{x-b}\right )$	Factor out $C$ from fraction
	$\displaystyle = C\lim \limits _{x\rightarrow b} \left (\frac {f(x) - f(b)}{x-b}\right )$	Factor out $C$ from limit using limit laws.

Again we used the limit law that you can move constant multipliers in and out of the limit expression freely on the last line, which gives us that you can do the same thing for derivatives!

We can record both these results up in the following rule:

Sum/Difference Rule Let $f(x)$ and $g(x)$ be differentiable functions, with derivatives $f'(x)$ and $g'(x)$ respectively. Then the derivative of $f(x) + g(x)$ is $f'(x) + g'(x)$ and the derivative of $f(x) - g(x)$ is $f'(x) - g'(x)$ . Furthermore, let $C$ be a real number, then the derivative of $Cf(x)$ is $Cf'(x)$ .

So as we have seen, the limit laws translate directly to derivatives in the case of sums, differences, and constant multiples. Indeed, you can break up the process of taking derivatives, handling each term of a function or expression independently just like you can when determining general limits. Moreover, the same is true for constant multipliers; any constant multiplier can be moved freely into, or out of, a derivative process... a fact that we will be using in a number of clever ways moving forward. It is worth a very important note here though that limits also break up nicely over multiplication and division, but as we will see in a future segment, this is not the case for derivatives. This means it is not simply a case that all derivatives laws are just copies of limit laws, despite how it may appear at first.