# Memorizing Formulas for Differentiation

Once upon a time, I memorized the following formula out of laziness.

Let $k(x)=\frac{f(x)^{g(x)}h(x)+i(x)}{j(x)}$. Then $k'(x)$ is as follows.

$k'(x)=\frac{j(x)(g(x)h(x)f(x)^{g(x)-1}f'(x)+f(x)^{g(x)}(h(x)log(f(x))g'(x)+h'(x)+i'(x))-j'(x)(h(x)f(x)^{g(x)}+i(x))}{j(x)^2}$

(as confirmed by wolframalpha)

This was because I did not want to bother with logarithmic functions or the chain rule to find the derivatives of functions such as $x^{sin(x)}$

After some considerable time and effort, I had managed to memorize the formula.

However, I had trouble actually applying this formula to tests for $f'(x)$, mostly because the formula is too long and complicated, and started to wonder if I had wasted my time and effort.

This suspicions were heightened when I made several mistakes while using this formula.

Would memorizing such a formula actually prove useful for tests?

• From a grader's standpoint, the grader might just skip your answer and not bother to give partial credit (if applicable) since the formula is incredibly convoluted and doesn't necessarily demonstrate your knowledge of the overall subject. Commented Feb 7, 2016 at 6:42
• @pyrazolam What's a grader? Do you mean a road grader? Or the teacher? I don't speak english. Commented Feb 7, 2016 at 6:43
• @pyrazolam I also only use this formula while trying to solve questions where you don't have to write down the solution, or multiple choice questions. I write down the normal solutions without using the formula when I know somebody is going to look at my solution. I am aware of the fact that it is highly likely that the teacher would not bother reading the extremely complex solution. Commented Feb 7, 2016 at 6:46
• The grader is a person who checks over your solution on the exam and decides if your solution is correct or not, then gives you points toward your exam score. So graders would either be the teachers themselves or a person the teacher has hired. Commented Feb 7, 2016 at 6:48
• I am sure many formulas are useful, but this particular one is not very useful. Using it is harder than not, at least in my opinion. Commented Feb 7, 2016 at 7:21

As already said in comments, it does not seem useful at all to memorize this. Using it does not prove that you know how to make it and there so many more important things to memorize in mathematics !

Let us do it simple $$k(x)=\frac{f(x)^{g(x)}h(x)+i(x)}{j(x)}=\frac{u(x)}{j(x)}$$ $$k'(x)=\frac{u'(x)j(x)-u(x)j'(x)}{j^2(x)}$$ $$u(x)=f(x)^{g(x)}h(x)+i(x)=t(x)h(x)+i(x)\implies u'(x)=t'(x)h(x)+t(x)h'(x)+i'(x)$$ $$t(x)=f(x)^{g(x)}\implies \log(t(x))=g(x)\log(f(x))$$ $$\frac{t'(x)}{t(x)}=g'(x)\log (f(x)) +g(x)\frac{ f'(x)}{f(x)}$$

By the way, if you had to program $k'(x)$ given the different functions $f(x)$, $g(x)$, $h(x)$, $i(x)$, $j(x)$, would you code the monster or just use pieces in the spirit of what I wrote ?

• The tone of this answer is strong. You appear to have a rather strong opinion on the subject at hand. It feels like I have personally insulted you(and if I did I am sorry). Thanks for your advice, Claude Leibovici. Commented Feb 7, 2016 at 7:52
• @Δαμον : you misunderstood what he wrote. and I personally think that there is nearly nothing to be memorized in maths, understanding is necessary and sufficient (the only one problem being not to forget several years after what we previously understood) Commented Feb 7, 2016 at 7:58
• @Δαμον. I am sorry if you consider that my answer had a strong tone. What I really wanted to say is that, in your studies and even later (at my age, I still learn everyday), there are so many important things to memorize that you overload your memory with a very small one that you will probably almost never use. About the interest of such complex formulae, I repeat : how would you code it ? Do you think that somebody else looking at the code will be able to understand it and maintain it ? Just be sure that I wanted to be positive. Cheers. Commented Feb 7, 2016 at 8:01
• @ClaudeLeibovici Actually, it was not a complaint. I did not mind your strong tone. I was just commenting that it seemed that you did. I am willing to listen to the advice of somebody who is far more experienced than me. If you did not know, I am 14 and undoubtedly one of the youngest people who use this site. Commented Feb 7, 2016 at 9:26
• @user1952009 While I think that objectively there might me nothing to be memorized, the reason I memorize these formula is for optimal performance on tests. The tests I normally watch require lots of calculations sometimes, and memorizing this formula often makes for a quicker solution. But I do know and understand how we got these formula. Commented Feb 7, 2016 at 9:32

Memorizing the formula is not much useful according to me. Instead, I would say, memorize the formula for chain rule and apply it to derive the formula in the exam. It is true that sometimes it might take more time, but it is sure to work and doesn't go wrong. Also, the problem will be quite simplified sometimes when you start by the fundamentals. I took many reputed tests and I speak from the experience ( I wrote IITJEE-2012 and CAT-2016 in India, and got 150-odd national rank in both ,if you know what I mean)

• Nice to see you again. Thanks for your answer. Did any of the links that I provided for Latex prove any help? Commented Feb 7, 2016 at 7:21
• Ya they did. Thanks a lot for that Commented Feb 7, 2016 at 8:29