2.6 Basic Combination Rules

Constant & constant multiple

The derivative of a constant is 0. A constant multiple pulls out: d/dx[k·f] = k·f'.

Sum & difference

d/dx[f ± g] = f' ± g' — differentiate term by term. So d/dx[x³ − 4x + 7] = 3x² − 4.

Key terms

Constant rule

d/dx[c] = 0.

Constant multiple rule

d/dx[k·f] = k·f'.

Sum/difference rule

d/dx[f ± g] = f' ± g'.

Term by term

Differentiate each term of a polynomial separately.

Example

d/dx[5x²] = 10x.

Try an AP-style question

d/dx (4x³ − 5x + 9) = ?

1. 12x² − 5
2. 4x² − 5
3. 12x² − 5 + 9
4. 12x³ − 5x