Let f and g be differentiable everywhere. If g is the inverse of f and g(-2)=5 and f'(5)=-1/2, then g'(2)=?
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To find g'(2), we can use the formula for the derivative of the inverse function:
g'(x) = 1 / f'(g(x))
Given that g(-2) = 5, we can find the value of x for g'(2) by solving for x in the equation g(x) = 2.
First, we find g(2):
g(2) = g(g(-2))
Since g(-2) = 5, g(2) = g(5)
Since g is the inverse of f, g(5) is the same as f'(5). So, g(2) = f'(5).
Now, substitute this into the original formula for g'(x):
g'(2) = 1 / f'(g(2))
g'(2) = 1 / f'(f'(5))
Finally, since f'(5) = -1/2, we can substitute this value:
g'(2) = 1 / (-1/2)
To divide by a fraction, we can multiply by its reciprocal:
g'(2) = 1 * (-2/1)
g'(2) = -2
Therefore, g'(2) = -2.