자연계열 2번

2-1>$\displaystyle 1=\frac d{dx}(x)=\frac d{dx}(\sin y)=\cos y\frac d{dx}(y)=\cos(g^{-1}(x))\frac d{dx}(g^{-1}(x))=\sqrt{1-x^2}\frac d{dx}(g^{-1}(x))$이므로 $\displaystyle \frac d{dx}(g^{-1}(x))=\frac1{\sqrt{1-x^2}}$이다.

2-2>$\displaystyle h^{\prime}(x)=(g^{-1})^{\prime}\left(\frac12x\right)\cdot\frac12=\frac1{\sqrt{1-\frac{x^2}4}}\cdot\frac12=\frac1{\sqrt{4-x^2}}$

2-3>$\displaystyle h^{\prime}(0)=\frac12$$\displaystyle h(0)=g^{-1}(0)=0$$\displaystyle y=\frac12x$