Suppose f and g are inverses, and (1,5) is a point on the graph of f. If f'(1) = -3, what, if anything, does that tell us about g'(1).
To find out what f'(1) tells us about g'(1), we need to understand the relationship between derivatives and inverses.
When two functions, f and g, are inverses of each other, their compositions result in the identity function. In other words, when we compose f and g, we get:
(f ∘ g)(x) = x
Similarly,
(g ∘ f)(x) = x
Now, let's differentiate both sides of the equation (f ∘ g)(x) = x using the chain rule. The chain rule states that if we have a composition of two functions, the derivative with respect to x is given by the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function with respect to x.
Differentiating both sides:
d/dx[(f ∘ g)(x)] = d/dx[x]
Applying the chain rule:
f'(g(x)) * g'(x) = 1
Since f and g are inverses, f(g(x)) = x, which means g(x) is equal to y-coordinate of the point (1,5), which is 5. Therefore:
f'(5) * g'(x) = 1
Now, we know that (f ∘ g)(x) = x, which means (f ∘ g)(5) = 5. Since f'(5) is given as -3, we have:
-3 * g'(x) = 1
Simplifying:
g'(x) = -1/3
So, f'(1) = -3 tells us that g'(x) = -1/3. Hence, the derivative of g at x = 1 is -1/3.