Question

Find the derivative of the inverse at a of f(x)=2x^3+3x^2+7x+4, a=4

Answer 1

check if a=4 is a root of f(x). If it is not, then the derivative of the inverse of f(x) at x=a is 1/f'(a)

Answer 2

\[
\frac{d}{dx}\frac{1}{f(x)} = -\frac{f'(x)}{[f(x)]^2}
\]

Answer 3

\(\frac{df}{dx}=6x^2+6x+7\\\frac{df}{dx}(4)=96+24+7=127\\\frac{df^{-1}}{dx}(4)=\frac{dx}{df}(4)=\frac1{\frac{df}{dx}(4)}=\frac1{127}\)

Answer 4

The answer key says it should result in 1/7. I am confused.

Answer 5

A function and its inverse have reciprocal slopes at all points, i.e. \(f'=\frac1{F'}\) and \(F'=\frac1{f'}\) where \(F=f^{-1}\). We find \(f'(x)=6x^2+6x+7\); evaluating at \(x=4\) yields \(f'(4)=6(4)^2+6(4)+7=96+24+7=127\). Our inverse's derivative is then \(F'(4)=\frac1{f'(4)}=\frac1{127}\).