Question

HELP!!!! Given that f and g are differentiable and that f(g(x)) = x. Also, f(2) = 4, f(4) = −6, f′(2) = 7, and f′(4) = 11. Determine g′(4). Show work that leads to your conclusion.

Answer 1

To find g'(4), we used the chain rule, given that f(g(x)) = x, and determined that f'(g(x)) · g'(x) = 1. After finding the corresponding values for f' and g, we calculated that g'(4) = 1/7.

Step-by-step explanation:

To find g'(4), we will use the fact that f(g(x)) = x and apply the chain rule to take the derivative of both sides with respect to x.

Following the chain rule, the derivative of f(g(x)) with respect to x is f'(g(x)) times g'(x). Since f(g(x)) = x, its derivative is 1. Therefore, we have:

f'(g(x)) · g'(x) = 1

We need to find the value of g'(4). We know that f(4) = -6, so g(-6) must equal 4 because f and g are inverse functions. Then, since f(2) = 4, we have g(4) = 2.

Now, we will plug in g(4) into the derivative:

f'(g(4)) · g'(4) = 1

f'(2) · g'(4) = 1

Given that f'(2) = 7, we can solve for g'(4):

7 · g'(4) = 1

g'(4) = 1/7

Thus, g'(4) is equal to 1/7.