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

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To find f^-1'(2) for the function f(x)=x³+3x+6, first determine the value of x where f(x)=2 by solving the equation x³+3x+6=2. This simplifies to x³+3x+4=0, which can be solved to find the corresponding x value. Once x is identified, calculate the derivative f'(x) to find its value at that specific x. The derivative of the inverse function is then given by (f^-1)'(2) = 1/f'(x). This method effectively utilizes the properties of inverse functions and their derivatives.
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given that f(x)=x³+3x+6

i need to find f^-1'(2)
 
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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.
 

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