# Find $f'(a)$ if $f(x) = \frac {x^n - a^n}{x-a}$

I'll state the multiple choice question from my textbook below:

If $f(x) = \frac {x^n - a^n}{x - a}$ for some constant '$a$', then $f'(a)$ is

(A) $1$

(B) $0$

(C) does not exist

(D) $\frac 12$

Here's what I tried:

By using quotient rule we get:

$f'(x) = \frac {(n-1)x^n - nax^{n-1} + a^n}{(x-a)^2}$

Clearly,

$f'(a) = \frac 00$

And the correct choice according to my book is (C). Does getting $\frac 00$ for a particular value of the variable (in this case, for $x=a$) mean that the derivative of the function doesn't exist at that point?

Not satisfied by the answer I factorised the numerator of $f(x)$ and cancelled the factor $(x-a)$ as below:

$f(x) = \frac {(x-a)(x^{n-1} + x^{n-2}a + x^{n-3}a^2 +..........+ xa^{n-2} + a^{n-1})}{x-a}$

$\implies f(x) = x^{n-1} + x^{n-2}a + x^{n-3}a^2 +..........+ xa^{n-2} + a^{n-1}$

Differentiating with respect to $x$ we get,

$f'(x) = (n-1)x^{n-2} + (n-2)x^{n-3}a + (n-3)x^{n-4}a^2 +..........+ a^{n-2}$

$\implies f'(a) = \Big[(n-1) + (n-2) + (n-3) + .......... + 1\Big]a^{n-2}$

$\implies f'(a) = \frac {n(n-1)}2 a^{n-2}$

Now this is a completely different answer.

So what have I done wrong? Is there some problem with cancelling the factor $(x-a)$? Does it have something to with the continuity and differentiabilty of $f(x)$ at $a$? What if the function was $f(x) = \begin{cases} \frac {x^n - a^n}{x - a}, & \text{if$x \ne a$} \\ na^{n-1}, & \text{if$x = a$} \end{cases}$? How do I find the derivative at $x = a$ in this case?

For the first case the derivative $f'(a)$ doesn't exist since $f(x)$ is not defined for $x=a$ and the derivative is defined by

$$\lim_{x\to a} \frac{f(x)-\color{red}{f(a)}}{x-a}$$

and the existence of $f(a)$ is required.

In the second case we are allowed to apply the definition by limit.

• Does the way I cancelled the factor $(x - a)$ from $f(x)$ work for the second case? Apr 7, 2018 at 15:16
• @SamInuyashaANMF Yes of course, the cancellation is allowed, see also there for a discussion about that math.stackexchange.com/questions/2628911/…
– user
Apr 7, 2018 at 15:19
• Well, the cancellation is allowed when $x \ne a$ — it's basic algebra in those cases, but when $x = a$ you are replacing $0/0$ with a real number, and you need to make some sort of "removing discontinuities" argument there, and you are redefining $f$, if subtly. Apr 7, 2018 at 15:28
• @JonathanZ Yes and it is allowed because $x \neq a$.
– user
Apr 7, 2018 at 15:29
• @SamInuyashaANMF The given function $f(x) = \frac {x^n - a^n}{x-a}$ is not defined at $x=a$ thus we can immeditely conclude that it is not differentiable at $x=a$ thus the expression $f'(a)$ is meaninglless. We can only evaluate $f'(x)$ for $x\neq a$. For the second function we can evaluate the derivative directly by the definition.
– user
Apr 7, 2018 at 15:52

If the correct answer is "C" then they are being very technical. The function $f$, as written, is not defined at $x = a$. So there's no hope of evaluating the limit $\dfrac{f(x) - f(a)}{x-a}$ — you don't have a value to put in for $f(a)$.

You have realized that the not-being-defined discontinuity at $x = a$ is removable, and in the second half of your question you've "rewritten" $f$ in a way that removes the discontinuity. I think that gives the most insight into what's happening, but technically your re-written $f$ is a different function than the original $f$ — one of them is defined at $a$ and one of them isn't.

Our human minds want to tell us to "redefine $f$ at any removable discontinuity so that it is continuous there", but rigorously mathematically it doesn't happen unless it is explicitly stated. It is not explicitly stated in the textbook's question, so "C" is the technically correct answer.

Now, as Ennar points out, it is important that you learn that the domain is an essential part of the definition of a function, especially when you are first starting to work with functions in calculus. Also, not explicitly giving the domain of a function happens a lot in math, so you have to get good at dealing with situations like this, and in a situation like this the technical answer is the correct answer.

As you go on and develop more tools and better math language you should deal less with multiple choice questions and more with ones that ask "Describe what happens if you want to differentiate the function $\dfrac{x^n-a^n}{x-a}$ at $x=a$.", and then it will be appropriate to do what you did in the second half of your question.

• I'm not sure I agree with the tone when you say "they are being very technical". We struggle to teach students that the domain of a function is essential part of it's definition, while you are making it sound it's just on a whim. $a$ is not in the domain of $f$ and the story ends there for this particular question. Adding $a$ to the domain in a continuous manner is a separate problem. Apr 7, 2018 at 15:29

The second way is correct. The problem with the first way is that the definition of $f$ doesn't hold at $a$. Presumably, the text means for the definition of $f$ to be extended to $a$ by continuity. Then you would take the limit of the difference quotient as $x\to a.$ When you get $\frac{0}0{}$ as a limit, you can't conclude that the limit doesn't exist. You'll learn about these "indeterminate forms" shortly, if you haven't already.

• Then is the book wrong to give the answer as choice (C)? Apr 7, 2018 at 15:19
• @Sam why? your $f$ isnt continuous so derivative does not exist. But yes this continuity is removable Apr 7, 2018 at 15:24
• See the answer given by JohnathanZ. To some extent, this is a matter of opinion, but I agree with him. The book is technically correct, but I think this is a poor way to ask a question of a beginner. Apr 7, 2018 at 15:25