If g is the inverse function of f, and f'(x) = 1/(1 + x5) and g(2) = 2, what is the value of g'(2)?
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Answer: If g is the inverse function of f, and f'(x) = 1/(1 + x5) and g(2) = 2, then the value of g'(2) = 33.
Let's understand the solution in detail.
The inverse g of a function f is given by the formula;
⇒ fog(x) = f(g(x)) = x
Now, when we differentiate both the sides in the above formula using the chain rule:
⇒ derivative of f(g(x)) = f'(g(x)).g'(x) = 1 (derivative of x w.r.t x is 1)
⇒ Now, we get g'(x) = 1/f'(g(x))
Also, using the given value of f'(x), and substituting g(x) in f'(x), we get:
⇒g'(x) = 1 + (g(x))5
Now, we substitute x = 2 in the above equation to get the required solution:
⇒g'(2) = 1 + (g(2))5 = 1 + 25 = 33 (given value of g(2) = 2)