Simplifying an advanced function question?

Question

This question caught me off guard..

For the following functions, simplify [f(x+h) - f(x) ] / [ h ]

a. f(x) = sqrt (2x)

There were two more, but once I understand how to solve these, I should be able to figure them out.

The main problem I have is when f(x) is instead f(x+h).....

Answer 1

You need to know Binomial Theorem.

[f(x+h) - f(x)] / h

= [sqrt(2x+2h) - sqrt(2x)] / h

= [{sqrt(2x) + (1/2)(2x)^(1/2 - 1)(2h) + ...} - sqrt(2x)] / h

= (1/2)(2x)^(1/2 - 1)(2h)/h as h-> 0

= (2x)^(-1/2)

Answer 2

You are studying differentiation.  The expression as given will not really simplify further:

[sqrt(2x + 2h) - sqrt(2x)]/h

What you are probably expected to do is to take h small and use a binomial expansion, to give

[sqrt(2x)+(1/2)(2x)^(-1/2)(2h)+(1/2)(-... - sqrt(2x)]/h

=1/sqrt(2x) - (2x)^(-3/2)h + ...

(using some very cavalier manipulation of infinite sums, which, by a miracle of mathematical analysis, actually works).