question about the limit $\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin(x)}{h}$

Let $\arcsin(x+h)=u$ and $\arcsin x=v$

$\displaystyle \implies (i) x+h=\sin u, x=\sin v$

and $\displaystyle (ii) -\frac\pi2\le u,v\le \frac\pi2$ (using the definition of principal value)

$\displaystyle\implies \cos u,\cos v\ge0$

$$\displaystyle\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin x}h=\lim_{u\to v}\frac{u-v}{\sin u-\sin v}$$

$$\displaystyle=\lim_{u\to v}\frac{u-v}{2\sin\frac{u-v}2\cos\frac{u+v}2}$$

$$\displaystyle=\lim_{u\to v}\frac{\frac{u-v}2}{\sin\frac{u-v}2} \cdot\frac1{\lim_{u\to v}\cos\frac{u+v}2}$$

$$\displaystyle=1\cdot\frac1{\cos v}=+\frac1{\sqrt{1-x^2}}$$

$$\text{Using }\arcsin p-\arcsin q=\arcsin(p\sqrt{1-q^2}-q\sqrt{1-p^2}),$$

$$\arcsin(x+h)-\arcsin x=\arcsin((x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2})$$

$$\displaystyle\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin x}h $$

$$=\lim_{h\to0}\frac{\arcsin((x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2})}h$$

$$=\lim_{h\to0}\frac{\arcsin((x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2})}{(x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2}}\cdot\lim_{h\to0}\frac{(x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2}}h$$

Putting $\displaystyle\arcsin((x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2})=v$ $\displaystyle\implies(x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2}=\sin v$ in the first limit,

$$\lim_{h\to0}\frac{\arcsin((x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2})}{(x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2}}=\lim_{v\to0}\frac v{\sin v}=1$$

$$\text{Again for the second limit, }\lim_{h\to0}\frac{(x+h)\sqrt{1-x^2}-x\sqrt{1-(x+h)^2}}h$$

$$=\lim_{h\to0}\frac{(x+h)^2(1-x^2)-x^2(1-(x+h)^2)}{h\{(x+h)\sqrt{1-x^2}+x\sqrt{1-(x+h)^2}\}}$$

$$=\lim_{h\to0}\frac{(x+h)^2-x^2}h\cdot\lim_{h\to0}\frac1{(x+h)\sqrt{1-x^2}+x\sqrt{1-(x+h)^2}}$$

$$=\lim_{h\to0}\frac{h(2x+h)}h\cdot\frac1{2x\sqrt{1-x^2}}$$

$$=\lim_{h\to0}(2x+h)\cdot\frac1{2x\sqrt{1-x^2}}\text{ as } h\ne0\text{ as } h\to0$$

$$=2x\cdot\frac1{2x\sqrt{1-x^2}}=\frac{1}{\sqrt{1-x^2}}$$