General question about math required to do david morins book

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RubinLicht
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So i don't have issues with most of his problems, but once in a while a question that requires a ridiculous math trick that i had no way of knowing comes up and i just wonder if this is something you pick up from various places, or can attain through a focused study of mathematics. the question that prompted me to post this was this:
2.png

and solution:
1.png


I get completely lost once i get to "In taking the derivative, the former dependence requires finding the value of theintegrand at the x0 limit...". and then i can follow the solution again once you get to "the solution to this is -> exponential"

The problem i had with this is that i immediately knew that the solution would be an exponential with some dependence on l in the exponent, however, there was simply no way for me to reach the solution because i had to use what ever math tricks he used here.

background: i can comfortably do approximations with taylor series and binomial theorem, so I'm not completely new to the world of approximations.

tldr i would like to know where to gain the math knowledge for these kind of tricks
 

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wrobel said:
this is not a trick this is a solution to the differential equation ##\rho'=-\rho##. There is a mathematical subject called "Ordinary Differential Equations"
"I get completely lost once i get to "In taking the derivative, the former dependence requires finding the value of theintegrand at the x0 limit...". and then i can follow thesolution again once you get to "the solution to this is -> exponential" "

Tldr I knew how to do that part, and you would know this is if you read my post again
 
wrobel said:
O, now I see. By the way have you ever heard about the Newton-Leibniz formula :)

https://www.encyclopediaofmath.org/index.php/Newton-Leibniz_formula

Yes but my understanding of it is very poor because we seemed to just skip over it in class. My friend also said to read about the fundamental theorem of calculus, so I'll do that when i get time and come back to this problem. Thanks!