Differentiate without know formula $[\arcsin x]' = \frac{1}{\sqrt{1-x^2}}$

Question

Is there any way how to get differentiate of $\arcsin x$ without memorize it?

$$[\arcsin x]' = \frac{1}{\sqrt{1-x^2}}$$

Answer 1

Let $t=\arcsin(x)$. We then have $x=\sin(t)$. Hence, we have $$\dfrac{dx}{dt} = \cos(t) = \sqrt{1-\sin^2(t)} = \sqrt{1-x^2} \implies \dfrac{dt}{dx} = \dfrac1{\sqrt{1-x^2}}$$

Answer 2

$$y = \arcsin x,$$ $$\sin y = x,$$ $$\cos y\frac{dy}{dx} = \frac{d\sin y}{dx} = 1,$$ $$\frac{dy}{dx} = \frac1{\cos y} = \frac1{\sqrt{1-\sin^2 y}}= \frac1{\sqrt{1-x^2}}.$$