Differentiation

Firstly when we see a question of Differentiation we jump into Differentiating it. Ok then carry on like that.... Then you would find a 0/0 form in the answer... So then comes the tricky part... Then use limits to solve further and and get the answer. Or else you can use some substitute for x....

$\begin{aligned} f(x) =& \sqrt{1-\sqrt{1-x^2}} \Rightarrow f(0) = 0\\ \ln(f(x)) = & \frac{1}{2}\ln(1-(1-x^2)^{\frac{1}{2}}) \\ \frac{f'(x)}{f(x)} =& -\frac{1}{2}\Big[ \frac{1}{1-(1-x^2)^{\frac{1}{2}}}\Big]\cdot \frac{x}{(1-x^2)^{\frac{1}{2}}}\\ f'(x) =& -\frac{1}{2}\Big[ \frac{(1-(1-x^2)^{\frac{1}{2}})^{\frac{1}{2}}}{1-(1-x^2)^{\frac{1}{2}}} \Big]\frac{x}{(1-x^2)^{\frac{1}{2}}}\\ f'(0) =& -\frac{1}{2}\lim_{x\to 0}\Big[ \frac{(1-(1-x^2)^{\frac{1}{2}})^{\frac{1}{2}}}{1-(1-x^2)^{\frac{1}{2}}} \Big]\cdot \lim_{x\to 0}\frac{x}{(1-x^2)^{\frac{1}{2}}} \end{aligned}$ The right side limit exists and evaluates to 0, therefore we are left with a 0/0 limit. $\begin{aligned} \lim_{x\to 0}\Big[ \frac{(1-(1-x^2)^{\frac{1}{2}})^{\frac{1}{2}}}{1-(1-x^2)^{\frac{1}{2}}} \Big] = \lim_{x\to 0} \frac{1}{1-(1-x^2)^{\frac{1}{2}}} \end{aligned}$ This limit obviously does not exist, so the derivative must not exist.