Answer:
C 1/2
Step-by-step explanation:
You have differentiable functions f and g that are inverses of each other with ...
- f(7) = -4
- f'(7) = 2
- g(-4) = 7
and you want to know g'(-4).
Solution
Consider the composition ...
f(g(x)) = x . . . . . . the functions are inverses of each other.
Differentiating with respect to x, we get ...
f'(g(x))·g'(x) = 1
g'(x) = 1/f'(g(x)) . . . . . . divide by the coefficient of g'(x)
For x = -4, this is ...
g'(-4) = 1/f'(g(-4)) = 1/f'(7) . . . . . use the given values
g'(-4) = 1/2
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Additional comment
In the attachment, you can consider the red line to be the tangent to f(x) at x=7, and the blue line to be tangent to g(x) at x=-4. The slope of the tangent, g'(-4), is 1/2, the reciprocal of the slope at f(7)=-4.
A function and its inverse are reflections of each other in the line y=x. That line is shown as dashed orange, and the points of interest are marked: (7, -4) and its inverse, (-4, 7).
