Question

Given that
`f(x) = x^2+x+1, f(-1) = 1` and
`g(x)` is the inverse of
`f(x)`, find
`g'(1)`

Include an explanation of how you found your answer.

Hint: You will probably want to use implicit differentiation.

Answer 1

-1

Explanation:

Note that by definition,

`f(g(x))=x \implies f'(g(x)) g'(x) = 1 \implies g'(x)=(1)/(f'(g(x)))`

Substituting x=1,

`g'(1)=(1)/(f'(g(1)))`

As g is the inverse of f,

`f(-1)=1 \implies g(1)=-1 \\ \\ \implies g'(1)=(1)/(f'(-1))`

By the power rule, f'(x)=2x+1, so f'(-1)=-1.

`\implies g'(1)=(1)/(-1)=-1`