Why can't the derivative of $e^{\ln x}$ be left as $\frac{e^{\ln x}}{x}$?

Question

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.

Answer 1

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:

• Why aren't the functions $f(x) = \frac{x-1}{x-1}$ and $f(x) = 1$ the same?

Answer 2

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.