# How can I explain this?

I am tutoring a kid who just started on calculus. Today he asked me if there is a way to expand this:
(a+b)^0.5

I said as far as I know, there is no way, but then the kid said:

f'(x^0.5)= ( (x+h)^0.5-x^0.5 ) / h = 0.5 x ^ -0.5;
(x + h)^0.5 = h(0.5 x ^ -0.5) + x^0.5;

and then, he proposed that x and h can be replaced by other variables, such as a and b, and the reason why the equation (a + b)^0.5 = b(0.5 a ^ -0.5) + a^0.5 is false is because h in the original equation approximated zero, and any term with h in it at the original derivative is removed, therefore, the "actual" way (a+b)^0.5 should be expanded is:

b(0.5 a ^ -0.5) + a^0.5 + c; where c is a series of functions in terms of a and b

I suspect that some of his logic is fallacious when doing the algebra to the derivative, because the equality is only true with h-->0, then again I am not very far into the logic of maths myself, so is there a way to demonstrate that this is wrong? I know the general formula for binomial expansion with integer exponents, but I don't have the proof, so maybe that will help if you can give it to me too.

andrewkirk
Homework Helper
Gold Member
There are two errors in the first line. The first one, which you identified, is that the limit as ##h\to 0## has not been taken, and the equation does not hold if the limit is not taken.

Secondly ##\frac{(x+h)^{0.5}-x^{0.5}}{h}\neq 0.5x^{0.5}##.

I suggest you ask him to justify his reasoning to you by working through the steps above one at a time. Working through and finding his own errors will be a more useful experience than having somebody else point them out for him.

One could use a Taylor series to expand the formula, but if he's just starting calculus then Taylor series may be a bit too advanced.