Use the definition of derivative to find $f'(x)$ for $f(x) = x^{1/2}$.

Question

This is an analysis question so we use the definition that the derivative of a function is, $$\lim_{x\to c} \frac{f(x)-f(c)}{x-c}$$ But I'm not really sure if I should just continue to say that $x=c$ for now as the second $x$ perhaps as if $f(x) = \sqrt c$. Obviously if we leave $x$ as is, $(f(x)-f(x)) = 0$ and the lim is $0$, but this is obviously not the case since $f'(x) = \frac{1}{2}\sqrt x$

Answer 1

Hint: $$\frac{ f(x)-f(c) }{ x-c }= \frac{\sqrt x - \sqrt c}{x-c}$$

and remark that: $$x-c=(\sqrt x - \sqrt c)(\sqrt x + \sqrt c)$$

Answer 2

Write $x$ as $c+h$, so you have $$\begin{align} f'(c)&=\lim_{h\to 0} \frac{\sqrt{c+h}-\sqrt{c}}{h} \\ &=\lim_{h\to 0}\frac{c+h-c}{(\sqrt{c+h}+\sqrt c)h} \\ &=\frac{1}{2\sqrt c} \end{align}$$