Given the function and derivative values in the table below, evaluate $\frac{d}{dx}f^{-1}(3)$
x: 1 2 3 4 5
f(x): 4 1 5 2 3
df/dx: 3 -1 4 0 -2
All I know is that the derivative of an inverse is $\frac{1}{f^\prime(f^{-1}(x))}$. Could anyone at least give me hints on how to use the table to my advantage? Thank you!
Best Answer
Note,
$$\frac{d}{dx}f^{-1}(x)=\frac{1}{f'(f^{-1}(x))}$$
Also we have $f(5)=3$, and thus $f^{-1}(3)=5$,
So $$\lim_{x\to 3}\frac{d}{dx}f^{-1}(x)=\frac{1}{f'(f^{-1}(3))}=\frac{1}{f'(5)}=-\frac{1}{2}$$ Sense $f'(5)=-2$