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)???
HELP i just need this one.
To find the value of g'(3), we need to use the inverse function theorem. According to the inverse function theorem, if f(x) and g(x) are inverse functions, then their derivatives satisfy the following relationship:
g'(x) = 1 / f'(g(x))
Given that g(3) = 5 and f'(5) = 4, we can substitute these values into the formula above to find the value of g'(3):
g'(3) = 1 / f'(g(3))
Let's start by finding f(3):
Since g(3) = 5, this means that f(5) = 3, as g(x) is the inverse of f(x).
Now, we can substitute f(5) = 3 and f'(5) = 4 into the formula:
g'(3) = 1 / f'(g(3)) = 1 / f'(5) = 1 / 4
So, the value of g'(3) is 1/4.