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

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The discussion focuses on finding the derivative of the inverse function, specifically f^-1'(2), for the function f(x) = x³ + 3x + 6. To determine f^-1'(2), one must first solve the equation x³ + 3x + 6 = 2 to find the corresponding x-value. The derivative of the inverse function is calculated using the formula (f^-1)'(y) = 1/f'(x), where f'(x) must be evaluated at the x-value that satisfies f(x) = 2.

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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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