Next suppose that interests are compounded continuously

Question

Suppose that a sum S₀ is deposited into a bank with an annual interest rate, denoted r. Denote S(t) to be the investment at time t, where t is the number of years since we've made the deposit. If the interest is compounded d times per years, then after t years, the investment value is given by

S(t) = S₀(1 + r/d)^dt.

(a) Next, suppose that interests are compounded continuously. Compute S(t) in that case by letting d tend to infinity in the previous equation

(1 + r/d)^dt = e^{In((1 + r/d)^dt)}

(b) We can alternatively derive this formula by solving a differential equation: Express S(t + 1/d) in terms of just S(t). With this expression, substitute it into the difference quotient

{S(t + 1/d) - S(t) } / {1/ d},

and let d tend to infinity, so show that S(t) satisfies the differential equation dS/dt = rS. Finally, solve this differential equation to confirm your result in part (a.).