If g(x) is the inverse of f(x), g(x)=f^-1(x), such that g(3)=5 and f'(5)=4 what is the value of g'(3)?????

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2. If g(x)is the inverse of f(x), g(x)=f^-1(x), such that f(3)=15,f(6)=3,f'(3)=-8,and f'(6)=-2. What is the value of g'(3)????????

Calculus--

To find the value of g'(3) in both cases, we can use the relationship between the derivatives of f(x) and g(x) when they are inverses of each other.

1. In the first scenario, we are given g(3) = 5 and f'(5) = 4. We want to find g'(3), which represents the derivative of g(x) evaluated at x = 3.

To find g'(3), we can use the inverse function theorem. According to the inverse function theorem, if f(x) and g(x) are inverses, then their derivatives are related by the formula:

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

So, if g(3) = 5, we know that g'(3) = 1 / f'(g(3)).

Substituting the given values, we get:

g'(3) = 1 / f'(5) = 1 / 4 = 1/4.

Therefore, the value of g'(3) is 1/4.

2. In the second scenario, we are given f(3) = 15, f(6) = 3, f'(3) = -8, and f'(6) = -2. We want to find g'(3), which represents the derivative of g(x) evaluated at x = 3.

Similar to the previous scenario, we can use the inverse function theorem to find g'(3). Since g(x) is the inverse of f(x), we have:

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

Given g(3) = 15, we can write:

g'(3) = 1 / f'(15).

To find f'(15), we need to determine the value of x such that f(x) = 15. From the given information, we know that f(3) = 15. Therefore, x = 3.

Hence, f'(15) = f'(3) = -8.

Substituting the values, we get:

g'(3) = 1 / f'(3) = 1 / (-8) = -1/8.

Therefore, the value of g'(3) is -1/8.