Derivative Rules: Constant, Sum, Difference, and Constant Multiple

A derivative is a limit, and limits of sums split apart: the limit of (f+g) is the limit of f plus the limit of g, as long as both limits exist. Since the difference quotient for f(x)+g(x) is just the difference quotient for f plus the difference quotient for g, taking the limit term by term gives (f+g)' = f' + g'. The same split works for subtraction. A constant multiple slides out of a limit entirely, since lim of c times h equals c times lim of h, which is why (c f)' = c f'. None of this needed a new idea - it is the old limit definition, applied to a sum instead of a single function.

This is why a polynomial like 5x⁴ - 3 sin(x) + 2e^x can be torn apart and handled piece by piece: differentiate 5x⁴ using the power rule, differentiate sin(x) and e^x using their own known derivatives, keep the minus sign and the constant multiples attached, and add the results. Nothing about combining terms requires a new rule.

The one place this linearity fails is multiplication. A product's difference quotient does not split into two separate pieces the way a sum's does - expanding f(x+h)g(x+h) - f(x)g(x) leaves cross terms mixing f and g, which is exactly why a genuine product rule is needed later. Keeping sum and difference rules separate from product and quotient rules is what makes the rest of this unit organized: constants and sums are free, products and quotients cost more work.