Constant and Sum Rules

Constant and Sum Rules

When differentiating expressions, two simple ideas are used often:

1. Constant rule: the derivative of a constant is 0.
2. Sum rule: the derivative of a sum is the sum of the derivatives.

These rules let you break a larger expression into smaller parts and differentiate term by term.

Method

1. Separate the expression into terms. Look for additions and subtractions.
2. Differentiate each term individually.
   • A constant term becomes 0.
   • A variable term is differentiated using the usual derivative rule for that term.
3. Keep the signs. If a term is subtracted, its derivative is also subtracted.
4. Simplify the result. Combine any like terms if possible and remove zero terms.

Example structure

For an expression like f(x) = 7 + x^3 - 4x, apply the rules term by term:

• derivative of 7 is 0
• derivative of x^3 is found separately
• derivative of -4x is found separately

Then add the results and simplify.

Check your work

A good check is to see whether every original term was handled once and whether all constants disappeared correctly. If your final answer still contains an unchanged constant, or if a sum was treated as one single piece, review the steps.