Inverse Function Derivatives

Core Sentence / 核心句

Inverse functions exchange inputs and outputs, so their derivatives are reciprocal slopes at corresponding points.

中文理解：反函数交换输入与输出，因此原函数和反函数在对应点的局部斜率互为倒数。

f (a) = b,

then

(f^{- 1})^{'} (b) = \frac{1}{f ^{'} ( a )} .

Equivalently:

(f^{- 1})^{'} (x) = \frac{1}{f ^{'} ( f ^{- 1} ( x ))} .

$f^{- 1} (x)$ is not $1/ f (x)$ .
Evaluate $f^{'}$ at the original input $a$ , not automatically at $b$ .
The formula requires the relevant inverse branch and a nonzero original derivative.

微积分BC第4次正课 · Inverse-Function Derivative
- corresponding points exchange coordinates and local slopes become reciprocals