We investigate functions which are solutions of the equation $x''=-2x^3$ and related ones. Solutions which satisfy also the initial conditions $x(0)=0, \: x'(0)=1$ and $x(0)=1, \: x'(0)=0$ are known as lemniscatic sine and cosine functions respectively. Taking our cue from the theory of elementary trigonometric functions, we give our own proof of the most remarkable properties and provide various formulae for relations between lemniscatic functions and their derivatives. Everywhere, if possible, the analogies with trigonometric functions are shown. Our main tool is the theory of Jacobi elliptic functions.