I want to solve an equation from first principles. The first principles equation is: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) – f(x)}{h}$$
$$f(x) = \frac{1}{\sqrt{x}} \text{ at } x= 1$$
Basically, I need to find the derivative, but I think I am getting my working out confused as the answer is $-1/2$.
Could you please show your working out so I can understand how to solve this? Thank you! 🙂
Also, I am having trouble understanding when using the same first principles formula how when $f(x) = 5$ the answer is $0$.
Thank you so much for your help. It is really appreciated!
Best Answer
$$ f'(3)=\lim_{h\to 0}{\displaystyle{1\over\sqrt{3+h}}-{1\over\sqrt{3}}\over h}= \lim_{h\to 0}{\displaystyle{1\over\sqrt{3+h}}-{1\over\sqrt{3}}\over h} {\displaystyle{1\over\sqrt{3+h}}+{1\over\sqrt{3}}\over\displaystyle{1\over\sqrt{3+h}}+{1\over\sqrt{3}}}= $$
$$ \lim_{h\to 0}{\displaystyle{1\over{3+h}}-{1\over{3}}\over h}{1\over\displaystyle{1\over\sqrt{3+h}}+{1\over\sqrt{3}}}= \cdots $$
Can you continue?