Answered

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Define the function
3
g(x) = x^3+ x
. If
f(x) = g^-1(x)
and
f(2) = 1
what is the value of
f'(2)
?

Sagot :

LammettHash

Given that

[tex]g(x) = x^3 + x[/tex]

the inverse [tex]g^{-1}(x)[/tex] is such that

[tex]g\left(g^{-1}(x)\right) = g^{-1}(x)^3 + g^{-1}(x) = x[/tex]

or

[tex]g\left(f(x)\right) = f(x)^3 + f(x) = x[/tex]

Differentiating both sides using the chain rule gives

[tex]3f(x)^2f'(x) + f'(x) = 1 \\\\ f'(x) \left(3f(x)^2+1\right) = 1 \\\\ f'(x) = \dfrac1{3f(x)^2+1}[/tex]

Then the derivative of f at 2 is

[tex]f'(2) = \dfrac1{3f(2)^2+1} = \boxed{\dfrac14}[/tex]

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