Given f(x) and g(x)=f^-1(x).
If f(1)=4 and f'(1)=-3, then find g'(4).
...No idea where to start with this. Please help?
Thanks much!
To find g'(4), we need to make use of the relationship between f(x) and g(x)=f^-1(x).
First, since f(1) = 4, we know that f^-1(4) = 1, because the inverse function "undoes" what the original function did.
Next, we want to find g'(4), which represents the derivative of g(x) at x=4. Since g(x) is the inverse of f(x), the derivative of g(x) with respect to x can be written as 1/f'(g(x)), as given by the inverse rule of derivatives.
Thus, we need to find f'(g(x)) at x=4. To do this, we first find g(4) using the fact that g(x) is the inverse of f(x).
Since g(4) = 1 (as mentioned earlier), we can rewrite f'(g(x)) = f'(1) = -3.
So g'(4) = 1/f'(g(4)) = 1/-3 = -1/3.
Therefore, g'(4) = -1/3.