if g(x)=f^(-1)(x), with f(5)=6 and g(7)=6, then g(6) equals 1/f(7) -1/f(7) -1/f(5) 1/f(5)

Answer:

D. $\frac{1}{f'(5)}$

Explanation:

Step1: Recall inverse - function derivative formula

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

Step2: Substitute $x = 6$ into the formula

We want to find $g'(6)$. By the formula $g'(x)=\frac{1}{f'(g(x))}$, when $x = 6$, we have $g'(6)=\frac{1}{f'(g(6))}$.

Step3: Use the given information

We know that $f(5)=6$, so $g(6) = 5$ (since $g(x)$ is the inverse of $f(x)$). Substituting $g(6)=5$ into the expression for $g'(6)$, we get $g'(6)=\frac{1}{f'(5)}$.