# Solve f^-1'(2): Find x for f(x)=x³+3x+6

• jamesbond007
In summary, to find f^-1'(2) given f(x) = x³+3x+6, we need to take the derivative of the inverse function f^-1(y) with respect to x, which can be expressed as (f^-1)'(f(x)) = 1/f'(x). We then solve for x in the equation f(x) = 2 and evaluate f'(x) at that value.
jamesbond007
given that f(x)=x³+3x+6

i need to find f^-1'(2)

jamesbond007 said:
given that f(x)=x³+3x+6

i need to find f^-1'(2)
Hi,
Derivatives of inverse functions for y=f(x)

$$y=f(x)\Rightarrow f^{-1}(y)=x=f^{-1}(f(x))$$ if we take derivatives of both sides dependent to x,

$$f'(x)[(f^{-1})'(f(x))]=1\Rightarrow(f^{-1})'(f(x))=\frac{1}{f'(x)}$$.

Of course, f'(x) has to be evaluated at the x such that f(x)= 2, so you need to start by solving $x^3+ 3x+ 6= 2$. Fortunately, that's very easy.

## 1. What does "Solve f^-1'(2)" mean?

The notation f^-1'(x) represents the inverse derivative of the function f(x). In other words, we are trying to find the value of x that gives a derivative of 2 for the inverse of the function f(x).

## 2. How do we find the inverse of a function?

To find the inverse of a function, we switch the x and y variables and solve for y. This will give us the inverse function, which can then be used to find the inverse derivative.

## 3. What is the process for finding the inverse derivative?

The process for finding the inverse derivative involves first finding the inverse of the function, then taking the derivative of the inverse function using the chain rule. Finally, we plug in the given value for the derivative and solve for x.

## 4. Can you provide an example of finding the inverse derivative?

Sure, for the function f(x) = x^2, the inverse function is f^-1(x) = √x. To find the inverse derivative at a point, say f^-1'(4), we first take the derivative of √x, which is 1/(2√x). Plugging in the given value of 4, we get f^-1'(4) = 1/(2√4) = 1/4. Therefore, the value of x for which the inverse derivative of f(x) equals 4 is 1/4.

## 5. Is there a specific method for solving for x in this type of problem?

Yes, there is a specific method for solving for x in this type of problem. After finding the inverse function and taking its derivative, we set the derivative equal to the given value and solve for x. This will give us the value of x that corresponds to the given derivative of the inverse function.

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