Question

True or False.

If $f(x) = 1/√x$ , then $f'(0)$ does not exist.

Answer 1

$f(x)=\dfrac {1}{\sqrt x}$ , now to check that derivative exist or not , we know that to check the derivative of function exist or not , we apply formula -

$f^{'}(x)= _{{lim\ h}{\to}0}\dfrac{f(x+h)-f(x)}{h}$

, now $_{limh\to0}$ $\dfrac {\dfrac {1}{\sqrt {x+h}}-\dfrac{1}{\sqrt x}}{h}$

$=_{limh\to0}$ $\dfrac {{ \dfrac {{\sqrt x}-{\sqrt{ x+h}}}{{\sqrt x}{\sqrt {x+h}}}}}{h}$

$=_{limh\to0}$ $\dfrac{x+{\sqrt x}{\sqrt {x+h}}-{\sqrt{x+h}{\sqrt {x}}-(x+h)}}{h{\sqrt{x}{\sqrt {x+h}}}({\sqrt{x}+\sqrt{x+h)}}}$ $=\dfrac {1}{x^{\dfrac {5}{2}}}$

hence we can clearly see that f'(0), of a function tends to infinity hence f'(0) does not exist .