Let f be the function defined by f(x) = x³ + x. If g(x) = f^-1(x) and g(2) = 1. What is the value of g'(2)?

Solution:

Given: Function f(x) = x³ + x

Differentiate w.r.t x 

f'(x) = 3x² + 1 --- (1)

Also g(x) = f^-1(x)

By definition of inverse functions, (gof)(x) = x and (fog)(x) = x

Consider (gof)(x) = x

Differentiate w.r.t x

g'[f(x)].f'(x) = 1 --- (2)

Now, to find g'(2) select ‘x’ such that f(x) = 2 and

By inspection, when x = 1, f(1) = 2 and f'(x) = 4.

Substituting x = 1 in (2),

g'[f(x)].f'(x) = 1

g'(2).4 = 1

g'(2) = 1/4

Therefore. the value of g'(2) is 1/4.

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Let f be the function defined by f(x) =x³ + x. If g(x) = f^-1(x) and g(2) = 1. What is the value of g'(2)?

Summary:

The function ‘f’ defined by f(x) = x³ + x. If g(x) = f^-1(x) and g(2) = 1, then value of g'(2) is 1/4.