Question

Let f be the function defined by f(x)=x^3+x. If g(x)=inverse function of x and g(2)=1, what is the value of g'(2)?

Answer 1

Let f(x) = y = x^3 + x

Since a cubic function is involved, I cannot write x(y) or g(x) explicitly.

y = 2 when x = 1. That is why g(2) = 1
dy/dx = 3x^2 + 1
When x = 1, which is when the inverse function g =2, then dy/dx = 4

The derivative of g(x) at that point is dx/dy = 1/(dy/dx) = 1/4