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.