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Yes, I know the answer is 1 because $e^{\ln x}$ is rewritten as $x^{\ln e}$, which is $x$, then multiplied by $\frac{1}{x}$ when taking the derivative of the inside function, resulting in the answer of $1$.

Just out of curiosity, why can't I just leave the answer as $\frac{e^{\ln x}}{x}$? I don't see anything wrong with that.

J W
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harold232
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  • What makes you think there is anything wrong with it? – Eric Wofsey Aug 13 '19 at 07:18
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    Of course you can, the question is why would you want to? – Hans Lundmark Aug 13 '19 at 07:18
  • I notice that every answer in my textbook does not leave the answer like that. It always uses this logarithmic property which I don't like memorizing – harold232 Aug 13 '19 at 07:19
  • because it's usually the normal way of solving any other derivative question. I just don't like doing this extra step of "rearranging" because it seems unusual and unnecessary to do it. I even used online derivative calculators and all seem to give the answer of 1 – harold232 Aug 13 '19 at 07:21
  • main question of this post is "can i leave it as (e^lnx)/x and is there anything wrong with it?" – harold232 Aug 13 '19 at 07:23
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    "this logarithmic property" $e^{\ln x}=x$ that you don't like memorizing is the definition of a logarithm. That's like leaving $\sqrt[3]{x^3}$ unsimplified instead of writing it as $x$, or $x/x$ instead of writing $1$. What would you say to someone who asked your question about $\sqrt[3]{x^3}$ or $x/x$? Would you agree with them if they said it "seems unusual and unnecessary" to simplify them? – anon Aug 13 '19 at 07:24
  • No because anyone can spot that easily. But with something like (e^lnx)/x, simplifying even further doesn't seem as eye-catching. – harold232 Aug 13 '19 at 07:29
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    It can be left like that of course, but it will give the teacher an impression that the student learnt chain rule in calculus earlier than logarithms. – Usermath Aug 13 '19 at 07:31
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    I can spot $e^{\ln x}$ as easily as I can spot $\frac xx$ or $\sqrt[3]{x^3}$. I have had a lot of practice, though. I think that's basically what it comes down to: How used are you to working with logarithms? – Arthur Aug 13 '19 at 07:32
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    "because $e^{\ln x}$ is rewritten as $x^{\ln e}$". I don't understand what you mean by that. We can see directly that $e^{\ln x} = x$, by the definition of $\ln x$. (If you understand logarithms, that should be effortless and automatic.) There is not an intermediate step where we rewrite $e^{\ln x}$ as $x^{\ln e}$. – littleO Aug 13 '19 at 07:33
  • You should learn to spot how to simplify $e^{\ln x}/x$ as easily as you would $\sqrt[3]{x^3}/x$. There are a lot of theorems, methods, formulas etc. you don't need to memorize in math, but you do need to remember definitions and what things mean - like $\ln x$ is the inverse of $\exp x$, which means composing these functions cancels out. – anon Aug 13 '19 at 07:34
  • I've learned logarithms, but very briefly as the last chapter, so I guess I forgot most of the rules. Either way, I just want to go with the flow and not stop working on derivatives and go back to logarithms. So in that regard, whenever I see something such as log or natural log as the exponent of e or a number, I rearrange it to simplify it? For example: e^ln4 becomes 4^lne (lne=1) and simifplies to 4? – harold232 Aug 13 '19 at 07:43
  • I would suggest that if you don't want to become familiar with, and fluent in the use of, the logarithmic property, you will struggle with a great deal of mathematics. It turns out that the exponential and logarithmic functions occur naturally in a wide variety of mathematical contexts. How, for example, will you define $x^y$ for arbitrary real numbers. So I think you have to leave matters of personal taste and affinity behind, and simply get used to it. I will then become easy, natural and familiar. – Mark Bennet Aug 13 '19 at 07:45
  • How will you define $x^y$ for arbitrary real numbers, and differentiate this as a function of $x$ or of $y$? – Mark Bennet Aug 13 '19 at 07:46
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    It is very silly to turn e^ln4 into 4^lne and then simplify to 4. As littleO pointed out, e^ln4 simplifies directly to 4, by definition, with no intermediate step. – anon Aug 13 '19 at 07:49
  • Most texts want you to simplify your answer. What do you think is simpler, writing $12,345,678-12,345,678$ or writing $0$? What about $1/(1/2)$ vs $2$? What about $f(f^{-1}(x))$ vs $x$? Yours is exactly like this last one. – MPW Aug 13 '19 at 08:11
  • I don't think it's the same as f(f^−1(x)) because anything multiplied by the inverse of itself cancels out. Though I can't say much because I forgot most of my logarithms. – harold232 Aug 13 '19 at 08:24
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    Yes take $ f(x) =e^x$ then $f^{-1}(x)=\ln(x)$ so $f(f^{-1}(x))=e^{\ln x} $ – kingW3 Aug 13 '19 at 08:28
  • I've improved the formatting of your mathematical expressions by using MathJax. For a basic tutorial on MathJax, see: https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference. That way, you can format mathematics in future questions yourself. – J W Aug 13 '19 at 09:13

2 Answers2

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When you can, it's usually a good idea to simplify. Think of it as becoming a habit because often, expressions such as $\sqrt[3]{x^3}$ might only arise as intermediate steps and for the rest of the calculations or work to come, you make life easier by simplifying when and where you can - as a general rule of thumb.

However, there is something I don't see pointed out yet in the comments. There is a (subtle?) difference between an equality such as $$\sqrt[3]{x^3} = x\tag{1}$$ and $$e^{\color{red}{\ln x}} = x \tag{2}$$ The first equality holds for all real numbers $x$ so there is no danger in replacing $\sqrt[3]{x^3}$ by $x$, in fact I would call it a good (mathematical) habit.

The second equality holds only for $x>0$ since for $x \le 0$, the logarithm $\color{red}{\ln x}$ in the left-hand side of the equality isn't defined. This is different for composing these functions the other way around, since $$\ln {e^x} = x \tag{3}$$ holds for all real numbers again.

For most exercises or practical purposes, you could say there's not much harm done in ignoring this. But if the domain of the function is relevant, then one advantage of leaving the derivative of $e^{\ln x}$ written as $\frac{e^{\ln x}}{x}$ is that you can still see it is only valid for $x>0$. Alternatively: you could simplify the expression but keep the condition on $x$.

Coming back to the comments on replacing $\frac{x}{x}$ by $1$, see also:

StackTD
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The whole idea of math is to simplify something so that it becomes usable. Knowing that the derivative is constant(1) is way more useful then having this expression which might not be constant, that you'd have to evaluate for each $x$.

In general my advice is don't skip anything in math because chances are it will either appear in future problems or it could help you solve future problems an easier way. Also building math part by part (block by block) makes math so much easier, from my experience I was okay in math until I've come back and relearnt all the stuff I've brushed off learning new stuff became so much easier because I didn't have to relearn math every time I started learning new lessons.

kingW3
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