ilajnarian05
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Given that [tex]f(x) = x^2+x+1, f(-1) = 1[/tex] and [tex]g(x)[/tex] is the inverse of [tex]f(x)[/tex], find [tex]g'(1)[/tex]

Include an explanation of how you found your answer.

Hint: You will probably want to use implicit differentiation.

Respuesta :

Medunno13

Answer:

-1

Step-by-step explanation:

Note that by definition,

[tex]f(g(x))=x \implies f'(g(x)) g'(x) = 1 \implies g'(x)=\frac{1}{f'(g(x))}[/tex]

Substituting x=1,

[tex]g'(1)=\frac{1}{f'(g(1))}[/tex]

As g is the inverse of f,

[tex]f(-1)=1 \implies g(1)=-1 \\ \\ \implies g'(1)=\frac{1}{f'(-1)}[/tex]

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

[tex]\implies g'(1)=\frac{1}{-1}=-1[/tex]

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