Derivative of $\sqrt{x}$ Using Symmetric Derivative Formula

Question

How do you find the derivative of $\sqrt{x}$ using the symmetric derivative formula?

$$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x-h)}{2h}. $$

I got stuck on trying to remove the h from the denominator.

Answer 1

I am assuming the symmetric derivative formula (also known as the symmetric difference) means that $$ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x-h)}{2h}. $$ Note that in your case you have to look at $$ \begin{split} L(h) &= \frac{\sqrt{x+h} - \sqrt{x-h}}{2h} \\ &= \frac{\sqrt{x+h} - \sqrt{x-h}}{2h} \times \frac{\sqrt{x+h} + \sqrt{x-h}}{\sqrt{x+h} + \sqrt{x-h}} \\ &= \frac{(x+h) - (x-h)} {2h \left(\sqrt{x+h} + \sqrt{x-h}\right)} \end{split} $$ Can you take it from here?