单变量微积分学习笔记：反函数求导法则（12）【6，9，11】

常用公式

\(\arcsin(x) = \frac{1}{\sqrt{1-x^2}}\)
\(\arccos(x) = -\frac{1}{\sqrt{1-x^2}}\)
\(\arctan(x) = \frac{1}{1+x^2}\)

证明

\(y = \arcsin(x)\)
\(\sin(y) = x\)
\(\cos(y)y' = 1\)
\(y' = \frac{1}{\cos(y)}\)
\(y' = \frac{1}{\sqrt{1-\sin^2(y)}}\)
\(y' = \frac{1}{\sqrt{1-x^2}}\)

---

\(y = \arccos(x)\)
\(\cos(y) = x\)
\(-\sin(y)y' = 1\)
\(y' = -\frac{1}{\sin(y)}\)
\(y' = -\frac{1}{\sqrt{1-\cos^2(y)}}\)
\(y' = -\frac{1}{\sqrt{1-x^2}}\)

---

\(y = \arctan(x)\)
\(\tan(y) = x\)
\(\sec^2(y)y' = 1\)
\(y' = \cos^2(y)\)

\(y' = \frac{1}{1+x^2}\)