Дифференцирование обратных функций

Slopes of a function and its mirror

• An inverse function 反函数 $f^{-1}$ undoes $f$: it reflects the graph across the line $y=x$.
• Reflecting swaps the roles of rise and run — so it flips the slope upside down.
• That gives a neat shortcut: you can find the inverse's derivative from the original's.
• Best of all, you often don't need a formula for $f^{-1}$ at all.

The reciprocal-slope rule

• At matching points, the slopes of $f$ and $f^{-1}$ are reciprocals 倒数.
• $$\big(f^{-1}\big)'(x)=\frac{1}{f'\!\big(f^{-1}(x)\big)}$$
• Read it as: the inverse's slope at $x$ is $1$ divided by $f$'s slope at the matching input.
• The matching input is $f^{-1}(x)$ — the point on $f$ that maps to $x$.

The inverse function is the mirror image of the function in the line y = x

Evaluate without inverting

• Usually you only need the derivative at one point, and you can skip finding $f^{-1}$.
• Say $f(2)=5$ and $f'(2)=3$. Then the point $(5,2)$ is on $f^{-1}$, so $f^{-1}(5)=2$.
• Apply the rule: $\big(f^{-1}\big)'(5)=\dfrac{1}{f'(2)}=\dfrac{1}{3}$.
• You never needed a formula — just the matching pair and one derivative value.

Why reciprocal? See the reflection

• Reflecting across $y=x$ turns a "rise of $3$ over run of $1$" into a "rise of $1$ over run of $3$."
• So a slope of $3$ becomes a slope of $\tfrac13$ — literally flipped.
• This is why $f$ and $f^{-1}$ have reciprocal slopes at corresponding points.
• (It also warns you: if $f'=0$ at a point, the inverse has a vertical tangent there — its slope is undefined.)