if g(x)=f^(-1)(x), with f(6)=2 and g(7)=2, then g(2) equals 1/f(7) f(6) f(7) 1/f(6)

Answer:

$\frac{1}{f'(6)}$

Explanation:

Step1: Recall inverse - function derivative formula

If $g(x)=f^{-1}(x)$, then $g'(x)=\frac{1}{f'(g(x))}$.

Step2: Substitute $x = 2$

We want to find $g'(2)$. By the formula $g'(2)=\frac{1}{f'(g(2))}$.

Step3: Use given information

We know that $g(x)=f^{-1}(x)$ and $f(6)=2$, so $g(2) = 6$.

Step4: Get the final result

Substituting $g(2)=6$ into $g'(2)=\frac{1}{f'(g(2))}$, we get $g'(2)=\frac{1}{f'(6)}$.