Help with a math question?

The answer is as follows:

Another way is to expand the terms first:
e^(5x) = 1 + 5x + (25/2)x^2 + (125/6)x^3 + (625/24)x^4 + (3125/120)x^5 + (15625/720)x^6 + ...
so that
d[e^(5x)]/dx = 5 + 25x + (125/2)x^2 + (625/6)x^3 + (3125/24)x^4 + ...

Remember the chain rule?  Let u = 5x and:

    d/dx u^n  =  (d/du u^n) * (du/dx)
                   =  n u^(n-1) * du/dx

Plug u=5x back in and note that du/dx = 5
   d/dx  (5x)^n = 5n * (5x)^(n-1)

The constant term in that sum on the right vanishes and the derivatives of the next 5 terms follow the above pattern.  When simplifying, you'll find that n/n! will reduce to 1/(n-1)! in each term, so that you end up 5 times the original series.

...which is just what you should expect from using the chain rule d/dx e^(5x) directly, without expanding the power series.