Verify the following steps yourself; they're pretty easy.
The derivative of the constant function f(x) = k is everywhere zero.
The derivative of the identity function f(x) = x is 1.
The derivative of x2 is 2x. (Use the product rule.)
The derivative of xn is nxn-1.
(Use the product rule and induction on n.)
A polynomial is a linear combination of powers of x. Hence we can differentiate term by term. Fold the exponents into the coefficients, decrementing the exponents along the way, and discard the constant term. For instance, the derivative of 3x3+7x2+39x+67 = 9x2+14x+39.