Question

Let f be a differentiable function such that f(-4)=12, f(9)=-4, f'(4)=-6, and f'(9)=3. The function g is differentiable and g(x)=finverse(x). What is the value of g'(-4)?

Answer 1

To find the value of g'(-4), we need to first find the derivative of the function g(x), which is the inverse of f(x).

Step-by-step explanation:

To find the value of g'(-4), we need to first find the derivative of the function g(x), which is the inverse of f(x). Since f(x) is differentiable, its inverse g(x) is also differentiable.

Let's start by finding the derivative of f(x). We are given that f'(-4)=12, f(9)=-4, f'(4)=-6, and f'(9)=3.

Now, let's find the derivative of g(x) using the formula for the derivative of an inverse function: g'(x) = 1/f'(g(x)).

Substituting x=-4 into the equation, we have: g'(-4) = 1/f'(g(-4)).

Since g(-4) is equal to -4 according to the definition of the inverse function, we have: g'(-4) = 1/f'(-4).

Now we can substitute f'(-4)=12 into the equation, giving us: g'(-4) = 1/12.

Therefore, the value of g'(-4) is 1/12.

Answer 2

My solution to the problem is as follows:

g'(4) = 1/f' (f^-1(-4)) = 1/f'( g(-4) )

By simplifying and calculating using your calculator, you can now arrive at the value of g'(4)

I hope my answer has come to your help. God bless and have a nice day ahead!