The derivative of an inverse:
Suppose that 𝑔=𝑓^−1 (f inverse) . If 𝑔(3)=2 , what is 𝑔′(3) ? Express your answer solely in terms of 𝑓 , not 𝑔 .
if g(3) = 2, then f(2) = 3
g'(3) = 1/f'(2)
thanks! I actually had that first but my formatting was wrong when entering it in.
To find the derivative of an inverse function, you can use the inverse function rule. Let's break down the steps to find 𝑔′(3) in terms of 𝑓.
Step 1: Find the value of 𝑓 at 𝑔(3).
Since 𝑔(3) = 2, we want to find the corresponding value of 𝑓(𝑔(3)) = 𝑓(2).
Step 2: Find the derivative of 𝑓(𝑥) with respect to 𝑥.
Let's assume 𝑓(𝑥) is a function of 𝑥. To find its derivative, use the standard methods such as the power rule, product rule, or chain rule.
Step 3: Find the derivative of the inverse function.
Apply the inverse function rule, which states that the derivative of the inverse function is equal to the reciprocal of the derivative of the original function evaluated at the corresponding value. In this case, 𝑔′(3) = 1 / 𝑓′(𝑔(3)).
So, the final answer for 𝑔′(3) solely in terms of 𝑓 is 1 / 𝑓′(𝑔(3)).