substitution method of letting u = x+1 is only for integrating, but you would only use that in the case where you have a non-linear function being integrated.
Letting u=x+1 is useful to see why the derivative of e^(x+1) is e^(x+1). It's especially useful for newcomers to calculus.
Yeh i agree. Its just a way of doing it and setting it out so you can trace your steps really, not needed once you're used to it but it helps to start with.
Also, you can write e^x+1 as (e^x) multiplied by (e^1), so it's equal to e(e^x). Since e is just a constant, it doesn't affect the differentiation, since differentiating kf(x) gives kf'(x) if k is a constant. Thus, differentiating it, it remains as ee^x=e^x+1
Also, you can write e^x+1 as (e^x) multiplied by (e^1), so it's equal to e(e^x). Since e is just a constant, it doesn't affect the differentiation, since differentiating kf(x) gives kf'(x) if k is a constant. Thus, differentiating it, it remains as ee^x=e^x+1
Out of all the above methods, this is the most useful one. By the time you've wrote down all these different substitutions it could of been solved by spotting this. IMO Its always best to spot what the answer should be or should look like, if this isn't possible, then proceed down the route of mathematical tricks.