In this pedagogical article, I explore a unified approach in obtaining the derivatives of functions and their inverses by adopting a guided self-discovery approach. I begin by finding the derivative of the exponential functions and the derivative of their inverses, the logarithmic functions. I extend this approach to generate formulae for the derivatives of inverse trigonometric functions. I strongly believe that this guided self-discovery approach should be adopted before obtaining the derivatives of the inverse functions using the chain rule or the general formula for the derivative of an inverse function. It is important to observe that this article is aimed at making a freshman or high school calculus class more entertaining, and so the algebraic proofs of most of the statements are reserved for future classes.