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 inverse functions.

1. g(x) = f^(-1)(x), g(3) = 5, and f'(5) = 4:

To find g'(3), we can use the inverse function derivative formula:

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

We know that g(3) = 5, so g'(3) = 1 / f'(g(3)).

Since f'(g(3)) = f'(5) = 4, we can substitute this value into the formula:

g'(3) = 1 / 4 = 0.25

Therefore, the value of g'(3) is 0.25.

2. g(x) = f^(-1)(x), f(3) = 15, f(6) = 3, f'(3) = -8, and f'(6) = -2:

To find g'(3), we can use the inverse function derivative formula:

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

We are given f(3) = 15, so g(15) = 3.

To find g'(3), we need to find f'(g(15)). Since f(6) = 3 and f'(6) = -2, we can use the derivative chain rule:

f'(g(15)) = f'(6) * g'(15)

f'(g(15)) = -2 * g'(15)

Since g(15) = 3, we can substitute these values into the formula:

f'(3) = -2 * g'(15)

We are given f'(3) = -8, so we get:

-8 = -2 * g'(15)

Dividing both sides by -2:

g'(15) = -8 / -2 = 4

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