AP Calculus AB / Unit 3: Differentiation: Composite, Implicit, and Inverse Functions
3.3 Differentiating Inverse Functions
Notes
If $g=f^{-1}$, then $$g'(b)=\frac{1}{f'(a)}\quad\text{where } f(a)=b$$
- Find the $a$ with $f(a)=b$ first; don't plug $b$ into $f'$.
Practice
$f(2)=7$ and $f'(2)=4$. If $g=f^{-1}$, then $g'(7)=$
$\displaystyle g'(7)=\frac{1}{f'(2)}$.
$f(x)=x^3+x$ and $g=f^{-1}$. Find $g'(2)$.
$f(1)=2$, $f'(1)=3+1=4$.
Graphically, the slopes of $f$ and $f^{-1}$ at matching points are:
Reflecting over $y=x$ swaps rise and run.
Ready for more? Take the Unit 3 test.