Let us start by letting f(x) = e^x and g(x) = lnx
And these functions are inverses of each other.
There is a fact associated with inverse functions that says that:
g'(x) = 1 / f '(g(x)) ..... so using this, we have
g'(x) = 1/ e^(g(x)) = 1/ e^(lnx) = 1/x (remember that f'(x) = e^x)
Putting this together, we have
(d/dx)[ ln(x) + C] = 1/x (the derivative of the constant "C" = 0)
Thus, taking the derivative of the anti-derivative ln(x) + C returns the function in the integrand, i.e., 1/x = x^(-1)
