The product rule: Y = u·v => dY/dX = d(u·v)/dx = u·dv + v·du
where u and v are two functions of X (u(X) and v(X), with the arguments dropped for simpler notation), and dv and du being the shorthand notation for du/dX and dv/dX.
    Example: Use the product rule to prove that MR = P for a PC firm and MR < P for a monopoly.
    Let u = P, where P (price) is a function of Q (quantity)
        v = Q
        Y = TR = u·v = P·Q.
    Using the product rule and substituting the example notation:
        Y = u·v =>  dY/dX = d(u·v)/dx = u·dv + v·du
        TR= P·Q => dTR/dQ = d(P·Q)/dQ = P·dQ/dQ + Q·dP/dQ
                                     = P + Q·dP/dQ
    Since dTR/dQ is MR, we have the final result: 
        (*) MR = P + Q·dP/dQ.
        (1) For a PC firm, dP/dQ = 0, so the second term in (*) drops out, leaving MR = P.
        (2) For a monopoly, dP/dQ < 0, so the second term in (*) is negative, making MR < P. QED

The quotient rule: Y = u/v => dY/dX = d(u/v)/dx = (v·du - u·dv)/v²
    Example: Use the quotient rule to prove that ATC falls when MC < ATC and ATC rises when MC > ATC.
    Let u = TC, where TC (total cost) is a function of Q (quantity)
        v = Q
        Y = ATC = u/v = TC/Q.
    Using the quotient rule and substituting the example notation:
         Y =  u/v =>  dY/dX  = d(u/v)/dx  = (v·du - u·dv)/v²
        ATC= TC/Q => dATC/dQ = d(TC/Q)/dQ = (Q·dTC/dQ - TC·dQ/dQ)/Q²
                                          = (Q·dTC/dQ - TC)/Q²
                                          = [(Q·dTC/dQ)/Q - (TC/Q)]/Q
                                          = [(dTC/dQ) - (TC/Q)]/Q
    Since dTC/dQ is MC, and TC/Q is ATC, we have the final result: 
        (*) dATC/dQ = [MC - ATC]/Q.
        (1) ATC falling => dATC/dQ < 0 => [MC-ATC] < 0 => MC < ATC.
        (2) ATC rising => dATC/dQ > 0 => [MC-ATC] > 0 => MC > ATC. QED