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)?

Functions form the backbone of many concepts in mathematics, which include calculus, algebra and geometry as well. The inverse of functions is also extensively used in these branches of mathematics. Let's solve a problem related to inverse functions on this page.

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.

Explanation:

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)

Hence, 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.