Find the derivative of each of the following functions, first by using the product rule, then by multiplying each function out and finding the derivative of the higher-order polynomial. Post both solutions along with explanations of the intermediate steps that demonstrate your understanding of the derivative.

1. LaTeX: \left(4x+6\right)\left(9x-5\right)
2. LaTeX: 4x^2\left(3x^3-2x+5x\right)
3. LaTeX: \left(3x^2\:-x\:+1\right)\left(7\:- \:x^6\right)
4. LaTeX: \left(h^3\:-\:5\right)\left(3h^2\:- \:5h\:-\:4\right)